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Theory and Applications of Categories, Vol. 16, No. 30, 2006, pp. 855–895.

A COHOMOLOGICAL DESCRIPTION OF CONNECTIONS ANDCURVATURE OVER POSETS

JOHN E. ROBERTS AND GIUSEPPE RUZZI

Abstract. What remains of a geometrical notion like that of a principal bundle whenthe base space is not a manifold but a coarse graining of it, like the poset formed by abase for the topology ordered under inclusion? Motivated by the search for a geometricalframework for developing gauge theories in algebraic quantum field theory, we give, inthe present paper, a first answer to this question. The notions of transition function,connection form and curvature form find a nice description in terms of cohomology, ingeneral non-Abelian, of a poset with values in a group G. Interpreting a 1–cocycle as aprincipal bundle, a connection turns out to be a 1–cochain associated in a suitable waywith this 1–cocycle; the curvature of a connection turns out to be its 2–coboundary. Weshow the existence of nonflat connections, and relate flat connections to homomorphismsof the fundamental group of the poset into G. We discuss holonomy and prove ananalogue of the Ambrose-Singer theorem.

1. Introduction

One of the outstanding problems of quantum field theory is to characterize gauge theoriesin terms of their structural properties. Naturally, as gauge theories have been successful indescribing elementary particle physics, there is a notion of a gauge theory in the frameworkof renormalized perturbation theory. Again, looking at theories on the lattice, there is awell defined notion of a lattice gauge theory.

This paper is a first step towards a formalism which adapts the basic notions of gaugetheories to the exigencies of algebraic quantum field theory. If successful, this should allowone to uncover structural features of gauge theories. Some earlier ideas in this directionmay be found in [17].

In mathematics, a gauge theory may be understood as a principal bundle over amanifold together with its associated vector bundles. For applications to physics, themanifold in question is spacetime but, in quantum field theory, spacetime does not enterdirectly as a differential manifold or even as a topological space. Instead, a suitable basefor the topology of spacetime is considered as a partially ordered set (poset), ordered underinclusion. This feature has to be taken into account to have a variant of gauge theorieswithin algebraic quantum field theory. To do this we adopt a cohomological approach.After all, a principal fibre bundle can be described in terms of its transition functions

We wish to thank Ezio Vasselli for discussions on the topic of this paper. We would like to thank thereferee of TAC for the fruitful comments on the preliminary version of this paper.

Received by the editors 2006-04-28 and, in revised form, 2006-11-28.Transmitted by Ronald Brown. Published on 2006-11-30.2000 Mathematics Subject Classification: 53C05, 18D05, 05E25.c© John E. Roberts and Giuseppe Ruzzi, 2006. Permission to copy for private use granted.

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856 JOHN E. ROBERTS AND GIUSEPPE RUZZI

and these form a 1–cocycle in Cech cohomology with values in a group G. We develophere a 1–cohomology of a poset with values in G and regard this as describing principalbundles over spacetimes. A different 1–cohomology has already proved useful in algebraicquantum field theory: a cohomology of the poset with values in a net of observablesdescribes the superselection sectors. The formalism developed here can be adapted tothis case.

We begin by explaining the notions of simplex, path and homotopy in the context ofposets showing that these notions behave in much the same way as their better knowntopological counterparts. We define the fundamental group of a path-connected posetwhich, in practice, coincides with the fundamental group of the underlying topologicalspace. We then explain the 1–cohomology of a poset with values in G linking it tohomotopy: the category of 1–cocycles is equivalent to the category of homomorphismsfrom the fundamental group to G.

Having defined principal bundles, we next introduce the appropriate notion of con-nection and curvature and investigate the set of connections on a principal bundle, thesebeing thus associated with a particular 1–cohomology. We discuss holonomy and prove aversion of the Ambrose-Singer Theorem.

We finally introduce the notion of gauge transformation and the action of the group ofgauge transformations on the set of connections of a principal bundle. We also relate flatconnections to homomorphisms from the fundamental group into G. We end by giving abrief outlook.

2. Homotopy of posets

In this section we analyze a simplicial set associated with a poset, having in mind twomain aims: first, we want to introduce the first homotopy group of posets because we willdiscuss cohomology later; secondly, we want to introduce the notion of inflating simpliceswhich is at the basis of the theory of connections. We will start by introducing symmetricsimplicial sets and defining their first homotopy group. The first homotopy group ofa poset K will be defined as that of a symmetric simplicial set Σ∗(K) associated withK. The inflationary structure of Σ∗(K) will be introduced and analyzed in the thirdsubsection. Throughout this section, we shall consider a poset K and denote its orderrelation by ≤. References for this section are [14, 20, 22].

2.1. Symmetric simplicial sets. A simplicial set Σ∗ is a graded set indexed by thenon-negative integers equipped with maps ∂i : Σn → Σn−1 and σi : Σn → Σn+1, with0 ≤ i ≤ n, satisfying the following relations

∂i∂j = ∂j∂i+1, i ≥ j; σiσj = σj+1σi, i ≤ j;

∂iσj = σj−1∂i, i < j; ∂jσj = ∂j+1σj = 1; ∂iσj = σj∂i−1, i > j + 1.(1)

The elements of Σn are called n–simplices ; the mappings ∂i and σi faces and degeneraciesrespectively. We shall denote: the compositions ∂i∂j, σiσj by the symbols ∂ij and σij,

A COHOMOLOGICAL DESCRIPTION OF CONNECTIONS AND CURVATURE 857

respectively; 0–simplices by the letter a; 1–simplices by b; 2–simplices by c and genericn–simplices by x. A simplicial set Σ∗ is said to be symmetric [8] if there are mappingsτi : Σn(K) → Σn(K) for n ≥ 1 and i ∈ {0, . . . n− 1}, satisfying the relations

τi τi = 1, τi τi−1τi = τi−1 τiτi−1, τj τi = τi τj i < j − 1;

∂jτi = τi−1 ∂j i > j, ∂i+1 = ∂iτi, ∂j τi = τi ∂j i < j − 1;

σjτi = τi+1σj i > j, σiτi = τi+1τiσi+1, τiσi = σi, σj τi = τi σj i < j − 1.

(2)

Furthermore, these relations imply that ∂i+1τi = ∂i and τiτi+1σi = σi+1τi. The mappingsτi define an action τ of the permutation group P(n+1) on Σn(K): it is enough to observethat relations (2) say that τi implements the transposition between the elements i andi + 1, and that transpositions generate the permutation group. Two n–simplices x and yare said to have the same orientation if there exists an even permutation σ of P(n + 1)such that y = τσx; they have a reverse orientation if there is an odd permutation σ ofP(n + 1) such that y = τσx. We denote by [x] the equivalence class of n–simplices whichhave the same orientation as x, and by [x] the equivalence class of n–simplices whoseorientation is the reverse of x. Notice that for any 0–simplex a we have [a] = {a}. For1–simplices we have [b] = {b}, while [b] = {b}, where b ≡ τ0b. Note that ∂0b = ∂1b, and∂1b = ∂0b. In the following we shall refer to b as the reverse of the 1–simplex b.

One of the aims of this section is to introduce the notion of the first homotopy groupfor a symmetric simplicial set associated with a poset. A purely combinatorial definitionof homotopy groups for arbitrary simplicial sets has been given by Kan [10] (see also[14]). This definition is very easy to handle in the case of simplicial sets satisfying theextension condition. Recall that a simplicial set Σ∗ satisfies the extension condition ifgiven 0 ≤ k ≤ n + 1 and x0, . . . , xk−1, xk+1, . . . xn+1 ∈ Σn, with n ≥ 1, such that

∂ixj = ∂j−1xi, i < j and i, j 6= k, (3)

then there is x ∈ Σn+1 such that ∂ix = xi for any i 6= k. Nevertheless, as we shall see in thenext subsection, the simplicial set associated with a poset does not satisfy this conditionin general. For an arbitrary symmetric simplicial set, however, there is an easy way tointroduce the first homotopy group. The procedure introduced in [20, 22] is easier tohandle than that of Kan which involves the group complex of a simplicial set. Moreover,in the case of the poset formed by a suitable base for the topology of a manifold, it leadsto a first homotopy group isomorphic to that of the manifold (see next subsection).

Here we recall definitions and results from [20, 22]. Given a0, a1 ∈ Σ0, a path p froma0 to a1, written p : a0 → a1, is a finite ordered sequence p = {bn, . . . , b1} of 1–simplicessatisfying the relations

∂1b1 = a0, ∂0bi = ∂1bi+1 for i ∈ {1, . . . , n− 1}, ∂0bn = a1. (4)

The starting point of p, written ∂1p, is the 0–simplex a0, while the endpoint of p, written∂0p, is the 0–simplex a1. The boundary of p is the ordered pair ∂p ≡ {∂0p, ∂1p}. A path


p is said to be a loop if ∂0p = ∂1p. Σ∗ is said to be pathwise connected whenever forany pair a0, a1 of 0–simplices there is a path p : a0 → a1. The set of paths is equippedwith the following operations. Given two paths p = {bn, . . . , b1} and q = {b′k, . . . b′1}, and∂0p = ∂1q, their composition q ∗ p is defined by

q ∗ p ≡ {b′k, . . . , b′1, bn, . . . , b1}.

The reverse of p is the pathp ≡ {b1, . . . , bn}.

Note that the reverse is involutive and the composition ∗ is associative. In particular notethat any path p = {bn, . . . , b1} can be also seen as the composition of its 1–simplices,i.e., p = bn ∗ · · · ∗ b1. An elementary deformation of a path p consists in replacing a1–simplex ∂1c of the path by a pair ∂0c, ∂2c, where c ∈ Σ2, or, conversely in replacing aconsecutive pair ∂0c, ∂2c of 1–simplices of p by a single 1–simplex ∂1c. Two paths with thesame endpoints are homotopic if they can be obtained from one another by a finite set ofelementary deformations. Homotopy defines an equivalence relation ∼ on the set of pathswith the same endpoints, which is compatible with reverse and composition, namely

p ∼ q ⇐⇒ p ∼ q,p ∼ q, p1 ∼ q1 ⇒ p1 ∗ p ∼ q1 ∗ q,

(5)

(the latter equation holds whenever the r.h.s. is defined). Furthermore, for any path p,the following relations hold:

p ∗ σ0∂1p ∼ p and p ∼ σ0∂0p ∗ p;p ∗ p ∼ σ0∂1p and σ0∂0p ∼ p ∗ p.

(6)

We prove the relation p ∗ p ∼ σ0∂1p the others follow in a similar fashion. It is clearlyenough to prove this relation for 1–simplices. So, given a 1–simplex b consider the 2–simplex τ0σ1b. Using relations (1) and (2) we have ∂0τ0σ1b = ∂1σ1b = b, ∂1τ0σ1b =∂0σ1b = σ0∂0b, and ∂2τ0σ1b = τ0∂2σ1b = τ0b = b. Hence, b ∗ b ∼ σ0∂0b.

Now, fix a ∈ Σ0, and define

π1(Σ∗, a) ≡ {p : a → a}/ ∼,

the quotient of the set of loops with endpoints a by the homotopy equivalence relation.Let [p] be the equivalence class associated with the loop p : a → a, and let

[p] · [q] = [p ∗ q], [p], [q] ∈ π1(Σ∗, a).

As a consequence of (5), (6), π1(Σ∗, a) with this composition rule is a group: the identityis the equivalence class [σ0a] associated with the degenerate 1–simplex σ0a; the inverseof [p] is the equivalence class [p] associated with the reverse p of p. π1(Σ∗, a) is thefirst homotopy group of Σ∗ based on a. If Σ∗ is pathwise connected, then π1(Σ∗, a) isisomorphic to π1(Σ∗, a1) for any a1 ∈ Σ0; this isomorphism class is the fundamental groupof Σ∗, written π1(Σ∗). If π1(Σ∗) is trivial, then Σ∗ is said to be simply connected.


2.2. Homotopy of a poset. Underlying cohomology is what is called the simplicialcategory ∆+ that can be realized in various ways. The simplest way is to take the objectsof ∆+ to be the finite positive ordinals, ∆n = {0, 1, . . . , n}, with n ≥ 0, and to take thearrows to be the monotone mappings. All these monotone mappings are compositionsof two particular simple types of mapping; the injective monotone mappings from oneordinal to the succeeding ordinal denoted di : (n − 1) → n, with i ∈ {0, 1, . . . , n}, anddefined as

di(k) ≡{

k k < i ,k + 1 otherwise ;

and the surjective monotone mappings from one ordinal to the preceding one denotedsi : (n + 1) → n, with i ∈ {0, 1, . . . , n}, and defined as

si(k) ≡{

k k ≤ i ,k − 1 otherwise .

The mappings di and si satisfy relations which are dual to the relations (1) satisfied by∂i and σi respectively.

We may also regard ∆n as a partially ordered set, namely as the set of its non-voidsubsets ordered under inclusion. We denote this poset by ∆n. Any map, in particulara monotone one, m : ∆n → ∆p induces, in an obvious way, an order-preserving map

of the partially ordered sets ∆n and ∆p, denoted by m. We can then define a singular

n–simplex of a poset K to be an order preserving map f : ∆n → K. We denote the setof singular n−simplices by Σn(K), and call the simplicial set of K the set Σ∗(K) of allsingular simplices. Note that a map m : ∆n → ∆p induces a map m∗ : Σp(K) → Σn(K),where m∗(f) ≡ f ◦ m with f ∈ Σp(K). In particular, the faces and degeneracies mappingare defined by

∂i : Σn(K) → Σn−1(K), where ∂i ≡ d∗i ,

σi : Σn(K) → Σn+1(K), where σi ≡ s∗i .

One can easily check that ∂i and σi satisfy the relations (1). A 0–simplex a is justan element of the poset. Inductively, for n ≥ 1, an n−simplex x is formed by n + 1(n − 1)−simplices ∂0x, . . . , ∂nx, whose faces are constrained by the relations (1), and bya 0–simplex |x| called the support of x such that |∂0x|, . . . , |∂nx| ≤ |x|. The orderedset {∂0x, . . . , ∂nx}, denoted ∂x, is called the boundary of x. We say that an n–simplex isdegenerate if it is of the form σiy for some (n−1)–simplex y and for some i ∈ {0, 1, . . . , n−1}. In particular we have |σiy| = |y|.

The next step is to show that Σ∗(K) is a symmetric simplicial set. Given a 1–simplexb, let τ0b be the 1–simplex defined by |τ0b| ≡ |b| and such that

∂0τ0b ≡ ∂1b, ∂1τ0b ≡ ∂0b. (7)

By induction for n ≥ 2: given an n–simplex x, for i ∈ {0, . . . n − 1}, let τix be then–simplex defined by |τix| ≡ |x| and

∂jτix ≡ τi−1∂jx i > j, ∂iτix ≡ ∂i+1x, ∂i+1τix ≡ ∂ix, ∂jτix ≡ τi∂jx i < j − 1. (8)


One can easily see that the mappings τi : Σn(K) → Σn(K), for n ≥ 1 and i ∈ {0, . . . , n−1}, satisfy the relations (2).

We now are concerned with the homotopy of a poset K, that is, the homotopy of thesymmetric simplicial set Σ∗(K) as defined in the previous subsection. To begin with weshow that Σ∗(K) does not fulfil the extension condition in general. To this end we recallthat a poset K is said to be directed whenever for any pair O1,O2 ∈ K, there is O3 ∈ Ksuch that O1,O2 ≤ O3.

We have the following

2.3. Lemma. Σ∗(K) is pathwise connected and satisfies the extension condition if, andonly if, K is directed.

Proof. (⇐) Clearly Σ∗(K) is pathwise connected. Let x0, . . . , xk−1, xk+1, . . . , xn+1 ∈Σn(K) satisfy the compatibility relations. As K is directed, we can find O ∈ K such thatO ≥ |x0|, . . . , |xk−1|, |xk+1|, . . . , |xn+1|. Let xk be the n–simplex defined by |xk| ≡ O and

∂ixk ≡ ∂k−1xi i < k, ∂kxk ≡ ∂kxk+1, ∂ixk ≡ ∂kxi+1, i > k.

Let w be the n + 1–simplex defined by |w| ≡ O and ∂iw ≡ xi. (⇒) Consider two 0–simplices a0, a1. Since Σ∗(K) is pathwise connected there is a path p : a0 → a1 of theform p = bn ∗ · · · ∗ b2 ∗ b1. Using the extension condition with respect to b2, b1, we can finda 2–simplex c such that ∂2c = b1 and ∂0c = b2. Hence, |c| ≥ a0 = ∂1b1, ∂0b2. Apply againthe extension condition with respect to b3, ∂1c. Then, there is a 2–simplex c1 such that∂0c1 = b3 and ∂2c = ∂1c. Hence, |c1| ≥ ∂11c = a0, ∂0b3. The proof follows by iterating thisprocedure until bn.

The condition on a poset to be directed is too restrictive. In fact, if K is directed,then Σ∗(K) has a contracting homotopy. Hence, K is simply connected [20, 22] (it couldbe easily seen that Σ∗(K) is simply connected also according to Kan’s definition of ho-motopy).

We will say that a poset K is pathwise connected, simply connected whenever Σ∗(K) ispathwise connected and simply connected, respectively. Furthermore, we define π1(K, a) ≡π1(Σ∗(K), a) and π1(K) ≡ π1(Σ∗(K)) and call them, respectively, the first homotopy groupof K, with base the 0–simplex a, and the fundamental group of K.

The link between the first homotopy group of a poset and the corresponding topologicalnotion can be achieved as follows. Let M be an arcwise connected manifold and let Kbe a base for the topology of M whose elements are arcwise and simply connected, opensubsets of M . Consider the poset formed by ordering K under inclusion. Then K ispathwise connected and π1(M) = π1(K), where π1(M) is the fundamental group of M[22, Theorem 2.18].

2.4. Inflationary structure. We deal with the inflationary structure of the simplicialset associated with a poset. We will describe the basic properties of this structure and usethem to give a definition of an inflationary structure for an abstract symmetric simplicialset. As we shall see later, theory of connections on a poset is based on this structure.


A 1−simplex b is said to be inflating whenever

∂1b ≤ ∂0b and |b| = ∂0b (9)

By induction for n ≥ 1: an n−simplex x is said to be inflating whenever all its (n− 1)–faces ∂0x, . . . , ∂nx are inflating (n − 1)–simplices and |x| = ∂123···(n−1)x. Any 0–simplexwill be regarded as inflating. We will denote the set of inflating n–simplices by Σinf

n (K).Given a monotone mapping m : ∆p → ∆n then m∗(x) = x ◦ m is inflating if x is. ThusΣinf∗ (K) is a simplicial subset of Σ∗(K).

Note that the 0–subsimplices of an inflating n–simplex form a totally ordered subsetof K. For instance, c is an inflating 2–simplex if, and only if,

∂11c = ∂12c ≤ ∂02c = ∂10c ≤ ∂00c = ∂01c = |c|.

Other properties of inflating simplices are shown in the following lemma.

2.5. Lemma. The following assertions hold.

(i) Σinf0 (K) = Σ0(K);

(ii) Given b, b′ ∈ Σinf1 (K). If ∂1b = ∂0b

′ and ∂0b = ∂1b′, then b = b′ = σ0a for some

0–simplex a.

(iii) Given b, b′ ∈ Σinf1 (K). If ∂0b = ∂1b

′, then there is c ∈ Σinf2 with ∂0c = b′ and ∂2c = b.

(iv) Given x, x′ ∈ Σinfn (K). If ∂ix = ∂ix

′ for i = 0, . . . , n, then x = x′.

(v) For any x ∈ Σn(K), there are b0, b1, . . . , bn ∈ Σinf1 (K) such that ∂1bk = ∂01···bk···nx for

0 ≤ k ≤ n, and ∂0b0 = ∂0b1 = · · · = ∂0bn.

Proof. (i) and (ii) are an easy consequence of the definition of inflating simplices. (iii)Define c by taking ∂2c = b, ∂0c = b′ and ∂1c ≡ (∂0b

′, ∂1b). (iv) is obvious. (v) Define |bk|and ∂0bk to be |x| and ∂1bk ≡ ∂012···bk···nx, for 0 ≤ k ≤ n.

The simplicial set Σinf∗ (K) does not, in general, satisfy the extension condition (3).

In fact, suppose there is a pair of 1–simplices b0, b1 ∈ Σinf1 (K) such that ∂0b0 = ∂0b1 but

neither ∂1b0 ≤ ∂1b1 nor ∂1b0 ≥ ∂1b1. Then, according to the definition of inflating simplex,an inflating 2–simplex c such that ∂0c = b0 and ∂1c = b1 does not exist. It is easy to seethat this is the case when K is a pathwise connected but not totally ordered poset.

The properties listed in Lemma 2.5 does not involve the notion of support of a sim-plex. This makes it possible to introduce an inflationary structure for abstract symmetricsimplicial sets, too.


2.6. Definition. A symmetric simplicial set Σ∗ has an inflationary structure, if thereis a simplicial subset Σ′

∗ of Σ∗ enjoying the properties (i)− (v) of the above lemma. Theelements of Σ′

∗ will be called inflating simplices.

Some observations on this definition are in order. Let Σ∗ be a symmetric simplicialset with an inflationary structure Σ′

∗. First, Property (ii) says that Σ′∗ is in general not

symmetric. Secondly, Property (iv) is a uniqueness condition: an inflating n–simplexis completely defined by its boundary. So sometimes, in the following, we will use thesymbol (a, a′) to denote the inflating 1–simplex b such that ∂0b = a and ∂1b = a′. Thirdly,consider two inflating 1–simplices of the form (a2, a1) and (a1, a0). Property (iii) impliesthat there is c ∈ Σ′

∗ such that ∂2c = (a1, a0) and ∂0c = (a2, a1). Since ∂1c ∈ Σ′∗, with

∂11c = a0, ∂01c = a2, then ∂1c = (a2, a0). Fourthly, Property (v) says that the set of0–subsimplices of an n–simplex admits an upper bound.

An important property of symmetric simplicial sets with an inflationary structure isshown in the following lemma.

2.7. Lemma. Let Σ∗ be an symmetric simplicial set with an inflationary structure Σ′∗.

Then writing a ≤ a′ whenever there is b ∈ Σ′1 with ∂1b = a and a′ ≤ ∂0b defines a partial

order relation on Σ0.

Proof. By (i) and (ii), the relation ≤ is reflexive and antisymmetric. Given a0, a1, a2 ∈Σ0 such that a0 ≤ a1 and a1 ≤ a2. By (iii) there is an inflating 2–simplex c with∂2c = (a1, a0), ∂0c = (a2, a1). Clearly, ∂1c = (a2, a0), hence a0 ≤ a2.

We denote the poset associated with Σ0 by Lemma 2.7 by the symbol K(Σ0).

An observation is in order. Assume that a poset K is given. Then it is clear thatK(Σ0(K)) = K. Conversely, assume that a symmetric simplicial set Σ∗ with an inflation-ary structure Σ′

∗ is given. Let Σ∗(K(Σ0)) be the simplicial set associated with the posetK(Σ0). Then, whilst Σinf

∗ (K(Σ0)) and Σ′∗ are simplicial equivalent, the same does not

hold for Σ∗(K(Σ0)) and Σ∗, because of the supports of the n–simplices of Σ∗(K(Σ0)).

Examples. Lemma 2.7 is the key to giving examples of an inflationary structure on asymmetric simplicial set. The basic strategy is to define a simplicial set Σinf

∗ using thesimplicial category ∆+ taking the partial ordering into account and another such Σ∗using the symmetric simplicial category and forgetting the partial ordering. Here aresome examples to illustrate this procedure, the first being just a variant of our basicexample. Let L be a poset and define Σinf

n (L) to be the set of order-preserving mappingsfrom {0, 1, 2, . . . , n} to L and Σn(L) to be an arbitrary mapping from {0, 1, 2, . . . , n} to L.Σinf∗ (L) is a simplicial subset of Σ∗(L) and it is easily checked that it defines an inflationary

structure on Σ∗(L). Note that Σinf∗ (L) is as defined previously but is now included in a

larger symmetric simplicial set.

Now let G be a partially ordered group and use a formulation in terms of homogeneoussimplices so that Σinf

∗ (G) and Σ∗(G) can be defined as if G were a partially ordered setbut we now have an action of G on the left compatible with the inclusion of Σinf

∗ (G) inΣ∗(G).


Let A be a partially ordered affine space and let Σn(A) be the set of affine mappingsfrom the standard simplex ∆n to A. Given such an affine map x write x ∈ Σinf

n (A) if thevertices of x are totally ordered a0 ≤ a1 ≤ · · · ≤ an. It is easily verified that we get aninflationary structure if A is directed.

Let U := {Ua} be a covering of a set X ordered under inclusion and define, as usual,Σ∗(U) taking a n–simplex x to be an ordered set of indices, {a0, a1, . . . , an} such that∩n

i=0Uaiis non-void. We say that x is inflating if a0 ≥ a1 ≥ · · · ≥ an and get in this

way a simplicial subset Σinf∗ (U). All the conditions for getting an inflationary structure

on Σ∗(U) are fulfilled except for (v). But this, too, is satisfied if we take U to be a baseof open sets for the topology of a topological space X.

The last example could be generalized by taking a poset P and defining Σn(P ) tobe the ordered sets {a0, a1, . . . , an} of elements of P such that there is an a ∈ P witha ≤ a0, a1, . . . , an.

3. Cohomology of posets

The present section deals with the, in general non-Abelian, cohomology of a pathwiseconnected poset K with values in a group G. The first part is devoted to explainingthe motivation for studying the non-Abelian cohomology of a poset and to defining ann–category. The general theory is developed in the second part: we introduce the set ofn–cochains, for n = 0, 1, 2, 3, the coboundary operator, and the cocycle identities up tothe 2nd-degree. In the last part we study the 1–cohomology, in some detail, relating itto the first homotopy group of a poset. It is worth stressing that definitions and results,presented in this section, on the cohomology of K rely only on the fact that Σ∗(K)is a symmetric simplicial set. Hence they admit an obvious generalization to abstractsymmetric simplicial sets.

3.1. Preliminaries. The cohomology of the poset K with values in an Abelian groupA, written additively, is the cohomology of the simplicial set Σ∗(K) with values in A. Tobe precise, one can define the set Cn(K, A) of n–cochains of K with values in A as theset of functions v : Σn(K) → A. The coboundary operator d defined by

dv(x) =n∑

k=0

(−1)k v(∂kx), x ∈ Σn(K),

is a mapping d : Cn(K, A) → Cn+1(K, A) satisfying the equation ddv = ι, where ι is thetrivial cochain. This allows one to define the n–cohomology groups. For a non-Abeliangroup G no choice of ordering gives the identity ddv = ι.

One motivation for studying the cohomology of a poset K with values in a non-Abeliangroup comes from the algebraic approach to quantum field theory. The leading idea ofthis approach [9] is that all the physical content of a quantum system is encoded in theobservable net, an inclusion preserving correspondence which associates to any open and


bounded region of Minkowski space the algebra generated by the observables measurablewithin that region. The collection of these regions forms a poset when ordered underinclusion. A 1–cocycle equation arises in studying charged sectors of the observable net:the charge transporters of sharply localized charges are 1–cocycles of the poset takingvalues in the group of unitary operators of the observable net [15]. The attempt toinclude more general charges in the framework of algebraic quantum field theory, chargesof electromagnetic type in particular, has led one to derive higher cocycles equations, upto the third degree [16, 17]. The difference, with respect to the Abelian case, is that a n–cocycle equation needs n composition laws. Thus in non-Abelian cohomology instead, forexample, of trying to take coefficients in a non-Abelian group the n–cocycles take valuesin an n–category associated with the group. The cocycles equations can be understoodas pasting together simplices, and, in fact, a n–cocycle can be seen as a representation inan n–category of the algebra of an oriented n–simplex [23].

Before trying to learn the notion of an n–category, it helps to recall that a categorycan be defined in two equivalent manners. One definition is based on the set of objectsand the corresponding set of arrows. However, it is possible to define a category referringonly to the set of arrows. Namely, a category is a set C, whose elements are called arrows,having a partial and associative composition law �, and such that any element of C hasleft and right �-units. This amounts to saying that (i) (f � g) �h is defined if, and only if,f � (g � h) is defined and they are equal; (ii) f � g � h is defined if, and only if, f � g andg � h are defined; (iii) any arrow g has a left and a right unit u and v, that is u � g = gand g � v = g. In this formulation the set of objects are the set of units.

An n–category is a set C with an ordered set of n partial composition laws. Thismeans that C is a category with respect to any such composition law �. Moreover, if ×and � are two such composition laws with � greater than ×, written × ≺ �, then:

1. every ×-unit is a �-unit;

2. ×-composition of �-units, when defined, leads to �-units;

3. the following relation, called the interchange law, holds:

(f × h) � (f1 × h1) = (f � f1)× (h � h1),

whenever the right hand side is defined.

An arrow f is said to be a k–arrow, for k ≤ n, if it is a unit for the k + 1 compositionlaw. To economize on brackets, from now on we adopt the convention that if × ≺ �,then a ×–composition law is to be evaluated before a �–composition. For example, theinterchange law reads

f × h � f1 × h1 = (f � f1)× (h � h1).

It is surprising that with this convention all the brackets disappear from the coboundaryequations (see below).


That an n–category is the right set of coefficients for a non-Abelian cohomology can beunderstood by borrowing from [3] the following observation. Assume that × is Abelian,that is, f × g equals g × f whenever the compositions are defined. Assume that �–unitsare ×–units. Let 1, 1′ be, respectively, a left and a right �–unit for f and g. By using theinterchange law we have

f � g = 1× f � g × 1′ = (1 � g)× (f � 1′) = g × f.

Hence � equals × and both composition laws are Abelian. Furthermore, if ? is a anothercomposition law such that × ≺ ? ≺ �, then × = ? = �.

3.2. Non-Abelian cohomology. Our approach to the non-Abelian cohomology of aposet will be based on n–categories. But it is worth pointing out that crossed complexescould be used instead (see [5]). In this approach cocycles turn out to be morphismsfrom a crossed complex associated with Σ∗(K) to a suitable target crossed complex. Thisapproach might be convenient for studying higher homotopy groups of K, but these areoutside the scope of the present paper.

Our first aim is to introduce an n–category associated with a group G to be usedas set of coefficients for the cohomology of the poset K. To this end, we draw on ageneral procedure [19] associating to an n–category C, satisfying suitable conditions, an(n+1)–category I(C). This construction allows one to define the (n+1)–coboundary of an–cochain in C as an (n + 1)–cochain in I(C), at least for n = 0, 1, 2. For the convenienceof the reader we review this construction in Appendix A.

Before starting to describe non-Abelian cohomology, we introduce some notation. Theelements of a group G will be indicated by Latin letters. The composition of two elementsg, h of G will be denoted by gh, and by e the identity of G. Let Inn(G) be the group ofinner automorphisms of G. We will use Greek letters to indicate the elements of Inn(G).By ατ we will denote the inner automorphism of G obtained by composing α with τ , thatis ατ(h) ≡ α(τ(h)) for any h ∈ G. The identity of this group, the identity automorphism,will be indicated by ι. Finally given g ∈ G, the equation

g α = τ g

means gα(h) = τ(h)g for any h ∈ G.

The categories nG . In degree 0, this is simply the group G considered as a set. Indegree 1 it is the category 1G having a single object, the group G, and as arrows theelements of the group. Composition of arrows is the composition in G. So we identifythis category with G. Observe that the arrows of 1G are invertible. By applying theprocedure provided in [19] (see Appendix A) we see that I(1G) is a 2–category, denotedby 2G, whose set of arrows is

2G ≡ {(g, τ) | g ∈ G, τ ∈ Inn(G)}, (10)


and whose composition laws are defined by

(g, τ)× (h, γ) ≡ (gτ(h), τγ),(g, τ) � (h, γ) ≡ (gh, γ), if σhγ = τ,

(11)

where σh is the inner automorphism associated with h. Some observations on 2G are inorder. Note that the composition × is always defined. Furthermore, the set of 1–arrows isthe set of those elements of 2G of the form (e, τ). Finally, all the 2–arrows are invertible.Now, in Appendix A we show that 2G satisfies all the properties required to define the3–category I(2G). However, it is not convenient to use this category as coefficients for thecohomology, but a category, denoted by 3G, which turns out to be isomorphic to I(2G)(Lemma A.2). The 3–category 3G is the set

3G ≡ {(g, τ, γ) | g ∈ Z(G), τ, γ ∈ Inn(G)}, (12)

where Z(G) is the centre of G, with the following three composition laws

(g, τ, γ)× (g′, τ ′, γ′) ≡ (gg′, ττ ′, γτγ′τ−1),(g, τ, γ) � (g′, τ ′, γ′) ≡ (gg′, τ ′, γγ′, ), if τ = γ′τ ′

(g, τ, γ) · (g′, τ ′, γ′) ≡ (gg′, τ, γ), if τ = τ ′, γ = γ′.(13)

Note that · is Abelian. The set of 1-arrows (3G)1 is the subset of elements of 3G of theform (e, γ, ι), where ι denotes the identity automorphism; 2–arrows (3G)2 are the elementsof 3G of the form (e, τ, γ). Finally, if G is Abelian, then × = � = · and the categories 2Gand 3G are nothing but the group G.

The set of n–cochains. The next goal is to define the set of n–cochains. Concerning 0–and 1–cochains nothing changes with respect to the Abelian case, i.e., 0– and 1–cochainsare, respectively functions v : Σ0(K) → G and u : Σ1(K) → G. A 2–cochain w is a pairof mappings (w1, w2), where wi : Σi(K) → (2G)i, for i = 1, 2 enjoying the relation

w2(c) � w1(∂1c) = w1(∂0c)× w1(∂2c) � w2(c), c ∈ Σ2(K). (14)

Thus associated with a 2–simplex c there is a 2–arrows w2(c) whose left and right �–unitsare computed from the boundary of c using w1. This equation and the definition of thecomposition laws in 2G entail that a 2–cochain w is of the form

w1(b) = (e, τb), b ∈ Σ1(K),

w2(c) = (v(c), τ∂1c), c ∈ Σ2(K),(15)

where v : Σ2(K) → G, τ : Σ1(K) → Inn(G) are mappings satisfying the equation1

v(c) τ∂1c = τ∂0c τ∂2c v(c), c ∈ Σ2(K). (16)

1Equation (16) means that v(c) τ∂1c(h) = τ∂0c(τ∂2c(h)) v(c) for any h ∈ G.


This can be easily shown. In fact, according to the definition of 2G a 2–cochain w is ofthe form w1(b) = (e, τb) for b ∈ Σ1(K), and w2(c) = (v(c), βc) for c ∈ Σ2(K). Now, thel.h.s. of equation (14) is defined if, and only if, τ∂1c = βc for any 2–simplex c. This factand equation (14) entail (16) and (15), completing the proof.

A 3–cochain x is 3–tuple (x1, x2, x3) where xi : Σ1(K) → (3G)i, for i = 1, 2, 3, satisfy-ing the following equations

x2(c) � x1(∂1c) = x1(∂0c)× x1(∂2c) � x2(c), (17)

for any 2–simplex c, and

x3(d) · x1(∂01d)× x2(∂3d) � x2(∂1d) = x2(∂0d)× x1(∂23d) � x2(∂2d) · x3(d), (18)

for any 3–simplex d. Proceeding as above, in Appendix A we show that these equationsand the composition laws of 3G entail that a 3–cochain x is of the form

x1(b) = (e, τb, ι), b ∈ Σ1(K),

x2(c) = (e, τ∂1c, γc), c ∈ Σ2(K),

x3(d) = (v(d), τ∂12d, γ∂0d γ∂1d), d ∈ Σ3(K),

(19)

where τ : Σ1(K) → Inn(G), v : Σ3(K) → Z(G), while γ : Σ2(K) → Inn(G) is themapping defined as

γc ≡ τ∂0c τ∂2c τ−1∂1c, c ∈ Σ2(K). (20)

This concludes the definition of the set of cochains. We will denote the set of n–cochainsof K, for n = 0, 1, 2, 3, by Cn(K,G).

Just a comment on the definition of 1–cochains: unlike the usual cohomological theo-ries 1–cochains are neither required to be invariant under oriented equivalence of simplicesnor to act trivially on degenerate simplices. However, as we shall see later, 1–cocyclesand connections fulfil these properties.

The coboundary and the cocycle identities. The next goal is to define thecoboundary operator d. Given a 0–cochain v, then

dv(b) ≡ v(∂0b) v(∂1b)−1, b ∈ Σ1(K). (21)

Given a 1-cochain u, then

(du)1(b) ≡ (e, ad(u(b))), b ∈ Σ1(K),

(du)2(c) ≡ (wu(c), ad(u(∂1c))), c ∈ Σ2(K),(22)

where wu is the mapping from Σ2(K) to G defined as

wu(c) ≡ u(∂0c) u(∂2c) u(∂1c)−1, c ∈ Σ2(K), (23)


while ad(·) means the adjoint action, that is, for instance, ad(u(∂1c)) is the inner auto-morphism associated with u(∂1c). Finally, given a 2-cochain w of the form (15), then

(dw)1(b) ≡(e, τb, ι

), b ∈ Σ1(K),

(dw)2(c) ≡(e, τ∂1c, γc

), c ∈ Σ2(K),

(dw)3(d) ≡(xw(d), τ∂12d, γ∂0d γ∂2d

), d ∈ Σ3(K),

(24)

where γ is the function from Σ2(K) to Inn(G) defined by τ as in (20), and xw is themapping xw : Σ3(K) → Z(G) defined as

xw(d) ≡ v(∂0d) v(∂2d)(τ∂01d(v(∂3d)) v(∂1d)

)−1(25)

for any 3–simplex d. Now, we call the coboundary operator d the mapping d : Cn(K, G) →Cn+1(K, G) defined for n = 0, 1, 2 by the equations (21), (22) and (24) respectively. Thisdefinition is well posed as shown by the following

3.3. Lemma. For n = 0, 1, 2, the coboundary operator d is a mapping d : Cn(K, G) →Cn+1(K, G), such that

ddv ∈ ((k + 2)G)k+1, v ∈ Ck(K, G)

for k = 0, 1.

Proof. The proof of the first part of the statement follows easily from the definition ofd, except that the function xw, as defined in (25), takes values in Z(G). Writing, forbrevity, vi for v(∂id) and τij for τ∂ij

, and using relations (1) and equation (16) we have

v0 v2 τ12 = v0 τ02 τ22 v2 = v0 τ10 τ22 v2 = τ00 τ20 τ23 v0 v2,

moreover

τ01(v3) v1 τ12 = τ01(v3) v1 τ11 = τ01(v3) τ01 τ21 v1

= τ01(v3 τ13) v1 = τ01(τ03 τ23 v3) v1 = τ01 τ03 τ23 τ01(v3) v1,

Hence both v(∂0d) v(∂2d) and τ∂01d(v(∂3d)) v(∂1d) intertwine from τ∂12d to τ∂01d τ∂03d τ∂23d.This entails that they differ only by an element of Z(G), proving that xw takes values inZ(G). Now, it is very easy to see that ddv ∈ (2G)1, for any 0–cochain v. So, let us provethat ddu ∈ (3G)2 for any 1–cochain u. Note that

(du)1(b) = (e, ad(u(b))), b ∈ Σ1(K)

(du)2(c) = (wu(c), ad(u(∂1c))), c ∈ Σ2(K),

where wu is defined by (23). Then the proof follows once we have shown that

wu(∂0d) wu(∂2d) = ad(u(∂01d))(wu(∂3d)

)wu(∂1d), (∗)


for any 3–simplex d. In fact, by (24) this identity entails that

(ddu)1(b) = (e, ad(u(b)), ι), b ∈ Σ1(K)

(ddu)2(c) = (e, ad(u(∂1c)), ad(wu(c))) c ∈ Σ2(K),

(ddu)3(d) = (e, ad(u(∂12d)), ad(wu(∂0d)wu(∂2d)

)), d ∈ Σ3(K).

So let us prove (∗). Given d ∈ Σ3(K) and using relations (1), we have

ad(u(∂01d))(wu(∂3d)

)wu(∂1d) = u(∂01d) wu(∂3d) u(∂01d)−1 wu(∂1d)

= u01 (u03 u23 u−113 ) u−1

01 (u01 u21 u−111 ) = u01 u03 u23 u−1

11

= u01 u03 u−102 u02 u23 u−1

11 = u00 u20 u−110 u02 u22 u−1

12

= wu(∂0d) wu(∂2d),

where we have used the notation introduced above. This completes the proof.

In words this lemma says that if v is a 0–cochain, then ddv is a 2–unit of 2G; if u is a1–cochain, then ddv is a 3–unit of 3G.

We now are in a position to define an n–cocycle and an n–coboundary.

3.4. Definition. For n = 0, 1, 2, an n–cochain v is said to be an n–cocycle whenever

dv ∈((n + 1)G

)n.

It is said to be an n–coboundary whenever

v ∈ d((n− 1)G

)(for n = 0 this means that v(a) = e for any 0–simplex a). We will denote the set ofn–cocycles by Zn(K, G), and the set of n–coboundaries by Bn(K, G).

Lemma 3.3 entails that Bn(K, G) ⊆ Zn(K, G) for n = 0, 1, 2. Although it is outsidethe scope of this paper, we note that this relation also holds for n = 3. One can checkthis assertion by using the 3–cocycle given in [16].

It is very easy now to derive the cocycle equations. A 0–cochain v is a 0–cocycle if

v(∂0b) = v(∂1b), b ∈ Σ1(K). (26)

A 1–cochain z is a 1–cocycle if

z(∂0c) z(∂2c) = z(∂1c), c ∈ Σ2(K). (27)

Let w = (w1, w2) be 2–cochain of the form w1(b) = (e, τb) for b ∈ Σ1(K), w2(c) =(v(c), τ∂1c) for c ∈ Σ2(K), where v and τ are mappings satisfying (16). Then w is a2–cocycle if

v(∂0d) v(∂2d) = τ∂01d

(v(∂3d)

)v(∂1d), d ∈ Σ3(K). (28)

In the following we shall mainly deal with 1–cohomology. Our purpose will be to show thatthe notion of 1–cocycle admits an interpretation as a principal bundle over a poset andthat this kind of bundle admits connections. The 2–coboundaries enter the game as thecurvature of connections. Since K is pathwise connected, it turns out that any 0–cocyclev is a constant function. Thus the 0–cohomology of K yields no useful information.


3.5. 1–Cohomology. This section is concerned with 1–cohomology. In the first part weintroduce the category of 1–cochains and the basic notions that will be used throughoutthis paper. The second part deals with 1–cocycles, where we shall derive some resultsconfirming the interpretation of a 1–cocycle as a principal bundle over a poset. In thelast part we discuss the connection between 1–cohomology and homotopy of posets.

The category of 1–cochains. Given a 1-cochain v ∈ C1(K, G), we can and willextend v from 1-simplices to paths by defining for p = {bn, . . . , b1}

v(p) ≡ v(bn) · · · v(b2) v(b1). (29)

3.6. Definition. Consider v, v1 ∈ C1(K, G). A morphism f from v1 to v is a functionf : Σ0(K) → G satisfying the equation

f∂0p v1(p) = v(p) f∂1p,

for all paths p. We denote the set of morphisms from v1 to v by (v1, v).

There is an obvious composition law between morphisms given by pointwise multipli-cation and this makes C1(K, G) into a category. The identity arrow 1v ∈ (v, v) takes theconstant value e, the identity of the group. Given a group homomorphism γ : G1 → Gand a morphism f ∈ (v1, v) of 1–cochains with values in G1 then γ ◦ v, defined as

(γ ◦ v)(b) ≡ γ(v(b)), b ∈ Σ1(K), (30)

is a 1–cochain with values in G, and γ ◦ f defined as

(γ ◦ f)a ≡ γ(fa), a ∈ Σ0(K), (31)

is a morphism of (γ ◦ v1, γ ◦ v). One checks at once that γ◦ is a functor from C1(K, G1) toC1(K, G), and that if γ is a group isomorphism, then γ◦ is an isomorphism of categories.

Note that f ∈ (v1, v) implies f−1 ∈ (v, v1), where f−1 here denotes the composition off with the inverse of G. We say that v1 and v are equivalent, written v1

∼= v, whenever(v1, v) is nonempty. Observe that a 1–cochain v is equivalent to the trivial 1–cochain ıif, and only if, it is a 1–coboundary. We will say that v ∈ C1(K, G) is reducible if thereexists a proper subgroup G1 ⊂ G and a 1–cochain v1 ∈ C1(K, G1) with ◦ v1 equivalentto v, where denotes the inclusion G1 ⊂ G. If v is not reducible it will be said to beirreducible.

A 1–cochain v is said to be path-independent whenever given a pair of paths p, q, then

∂p = ∂q ⇒ v(p) = v(q) . (32)

Of course, if v is path-independent then so is any equivalent 1–cochain. It is worthobserving that if γ is an injective homomorphism then v is path-independent if, and onlyif, γ ◦ v is path-independent.

3.7. Lemma. Any 1–cochain is path-independent if, and only if, it is a 1–coboundary.


Proof. Assume that v ∈ C1(K, G) is path-independent. Fix a 0–simplex a0. For any 0–simplex a, choose a path pa from a0 to a and define fa ≡ v(pa). As v is path-independent,for any 1–simplex b we have

v(b) f∂1b = v(b) v(p∂1b) = v(b ∗ p∂1b) = v(p∂0b) = f∂0b.

Hence v is a 1–coboundary, see (21) and Definition 3.4. The converse is obvious.

1–Cocycles as principal bundles. Recall that a 1–cocycle z ∈ Z1(K, G) is a mappingz : Σ1(K) → G satisfying the equation

z(∂0c) z(∂2c) = z(∂1c), c ∈ Σ2(K)

Some observations are in order. First, the trivial 1–cochain ı is a 1–cocycle (see Section3.1). So, from now on, we will refer to ı as the trivial 1-cocycle. Secondly, if z is a 1–cocycle then so is any equivalent 1–cochain. In fact, let v ∈ C1(K, G) and let f ∈ (v, z).Given a 2–simplex c we have

v(∂0c) v(∂2c) = f−1∂00c z(∂0c) f∂10c f−1

∂02c z(∂2c) f∂12c

= f−1∂00c z(∂0c) z(∂2c) f∂12c = f−1

∂00c z(∂1c) f∂12c

= f−1∂01c z(∂1c) f∂11c = v(∂1c),

where relations (1) have been used.

3.8. Lemma. Let γ : G1 → G be a group homomorphism. Given v ∈ C1(K, G1) considerγ ◦ v ∈ C1(K, G). Then: if v is a 1–cocycle, then γ ◦ v is a 1–cocycle; the converse holdsif γ is injective.

Proof. If v is a 1–cocycle, it is easy to see that γ ◦ v is a 1–cocycle too. Conversely,assume that γ is injective and that γ ◦ v is a 1–cocycle, then

γ(v(∂0c) v(∂2c)

)= γ ◦ v(∂0c) γ ◦ v(∂2c) = γ ◦ v(∂1c) = γ

(v(∂1c)

)for any 2–simplex c. Since γ is injective, v is a 1–cocycle.

Given a 1–cocycle z ∈ Z1(K, G), a cross section of z is a function s : Σ0(U) → G,where U is an open subset of K with respect to the Alexandroff topology 2, such that

z(b) s∂1b = s∂0b, b ∈ Σ1(U) . (33)

The cross section s is said to be global whenever U = K. A reason for the terminologycross section of a 1–cocycle is provided by the following

3.9. Lemma. A 1–cocycle is a 1–coboundary if, and only if, it admits a global cross section.

Proof. The proof follows straightforwardly from the definition of a global cross sectionand from the definition of a 1–coboundary.

2The Alexandroff topology of a poset K is a T0 topology in which a subset U of K is open whenevergiven O ∈ U and O1 ∈ K, with O ≤ O1, then O1 ∈ U .


3.10. Remark. Given a group G, it is very easy to define 1–coboundaries of the posetK with values in G. It is enough to assign an element sa ∈ G to any 0–simplex a and set

z(b) ≡ s∂0b s−1∂1b, b ∈ Σ1(K).

It is clear that z is a 1–cocycle. It is a 1–coboundary because the function s : Σ0(K) → Gis a global cross section of z. As we shall see in the next section, the existence of 1–cocycles, which are not 1–coboundaries, with values in a group G is equivalent to theexistence of nontrivial group homomorphisms from the first homotopy group of K into G.

We call the category of 1–cocycles with values in G, the full subcategory of C1(K, G)whose set of objects is Z1(K, G). We denote this category by the same symbol Z1(K, G)as used to denote the corresponding set of objects. It is worth observing that, given agroup homomorphism γ : G1 → G, by Lemma 3.8, the restriction of the functor γ◦ toZ1(K, G1) defines a functor from Z1(K, G1) into Z1(K, G).

We interpret 1–cocycles of Z1(K, G) as principal bundles over the poset K, having G asa structure group. It is very easy to see which notion corresponds to that of an associatedbundle in this framework. Assume that there is an action A : G×X 3 (g, x) → A(g, x) ∈X of G on a set X. Consider the group homomorphism α : G 3 g → αg ∈ Aut(X) definedby

αg(x) ≡ A(g, x), x ∈ X,

for any g ∈ G. Given a 1–cocycle z ∈ Z1(K, G), we call the 1–cocycle

α ◦ z ∈ Z1(K, Aut(X)), (34)

associated with z, where α◦ is the functor, associated with the group homomorphism α,from the category Z1(K, G) into Z1(K, Aut(X)).

Homotopy and 1–cohomology. The relation between the homotopy and the 1–cohomology of K has been established in [22]. Here we reformulate this result in thelanguage of categories. We begin by recalling some basic properties of 1–cocycles. First,any 1–cocycle z ∈ Z1(K, G) is invariant under homotopy. To be precise given a pair ofpaths p and q with the same endpoints, we have

p ∼ q ⇒ z(p) = z(q). (35)

Secondly, the following properties hold:

(a) z(p) = z(p)−1 , for any path p;(b) z(σ0a) = e , for any 0–simplex a.

(36)

Now in order to relate the homotopy of a poset to 1–cocycles, a preliminary definition isnecessary.

Fix a group S. Given a group G we denote the set of group homomorphisms from S


into G by H(S, G). For any pair σ, σ1 ∈ H(S, G) a morphism from σ1 to σ is an elementh of G such that

hσ1(g) = σ(g) h, g ∈ S. (37)

The set of morphisms from σ1 to σ is denoted by (σ1, σ) and there is an obvious com-position rule between morphisms yielding a category again denoted by H(S, G). Given agroup homomorphism γ : G1 → G, there is a functor γ◦ : H(S, G1) → H(S, G) defined as

γ ◦ σ ≡ γσ σ ∈ H(S, G1);γ ◦ h ≡ γ(h) h ∈ (σ, σ1), σ, σ1 ∈ H(S, G1).

(38)

When γ is a group isomorphism, then γ◦ is an isomorphism of categories, too. Similarly,let S1 be a group and let ρ : S1 → S be a group homomorphism. Then there is a functor◦ρ : H(S, G) → H(S1, G) defined by

σ ◦ ρ ≡ σρ σ ∈ H(S, G);h ◦ ρ ≡ h h ∈ (σ, σ1), σ, σ1 ∈ H(S, G).

(39)

When ρ is a group isomorphism, then ◦ρ is an isomorphism of categories, too.Now, fix a base 0–simplex a0 and consider the category H(π1(K, a0), G) associated

with the first homotopy group of the poset. Then

3.11. Proposition. Given a group G and any 0–simplex a0 the categories Z1(K, G) andH(π1(K, a0), G) are equivalent.

Proof. Let us start by defining a functor from Z1(K, G) to H(π1(K, a0), G). Givenz, z1 ∈ Z1(K, G) and f ∈ (z1, z), define

F (z)([p]) ≡ z(p), [p] ∈ π1(K, a0);

F (f) ≡ fa0 .

F (z) is well defined since 1–cocycles are homotopy invariant. Moreover, it is easy to seeby (36) that F (z) is a group homomorphism from π1(K, a0) into G. Note that

fa0 F (z1)([p]) = fa0 z1(p) = z(p) fa0 = F (z)([p]) fa0 ,

hence F (f) ∈ (F (z1), F (z)). So F is well defined and easily shown to be a covariantfunctor. To define a functor C in the other direction, let us choose a path pa from a0 toa, for any a ∈ Σ0(K). In particular we set pa0 = σ0(a0). Given σ ∈ H(π1(K, a0), G) andh ∈ (σ1, σ), define

C(σ)(b) ≡ σ([p∂0b ∗ b ∗ p∂1b]) , b ∈ Σ1(K);

C(h) ≡ c(h),

where c(h) : Σ0(K) → G is the constant function taking the value h for any a ∈ Σ0(K).It is easily seen that C is a covariant functor. Concerning the equivalence, note that

(F · C)(σ)([p]) = C(σ)(p) = σ([σ0(a0) ∗ p ∗ σ0(a0)]) = σ([p]),


and that (F · C)(h) = F (c(h)) = h. Hence F · C = idH(π1(K,a0),G). Conversely, given a1–simplex b we have

(C · F )(z)(b) = F (z)([p∂0b ∗ b ∗ p∂1b]) = z(p∂0b)−1 z(b) z(p∂1b),

and given a 0–simplex a we have (C · F )(f) = C(fa0) = c(fa0). Define u(z)a ≡ z(pa) fora ∈ Σ0(K). It can be easily seen that the mapping Z1(K, G) 3 z → u(z) defines a naturalisomorphism between C · F and idZ1(K,G). See details in [22].

Observe that the group homomorphism corresponding to the trivial 1–cocycle ı is thetrivial one, namely σ([p]) = e for any [p] ∈ π1(K, a0). Hence, a 1–cocycle of Z1(K, G) is a1–coboundary if, and only if, the corresponding group homomorphism F (z) is equivalentto the trivial one. In particular if K is simply connected, then Z1(K, G) = B1(K,G).

The existence of 1–cocycles, which are not 1–coboundaries, relies, in particular, on thefollowing corollary

3.12. Corollary. Let M be a nonempty, Hausdorff and arcwise connected topologicalspace which admits a base for the topology consisting of arcwise and simply connectedsubsets of M . Let K denote the poset formed by such a base ordered under inclusion ⊆.Then

H(π1(M, x0), G) ∼= H(π1(K, a0), G) ∼= Z1(K, G) ,

for any x0 ∈ M and a0 ∈ Σ0(K) with x0 ∈ a0, where ∼= means equivalence of categories.

Proof. π1(M, x0) is isomorphic to π1(K, a0) [22, Theorem 2.18] (see also Section 2). Asobserved at the beginning of this section, this entails that the categories H(π1(M, x0), G)and H(π1(K, a0), G) are isomorphic. Therefore the proof follows by Proposition 3.11.

Let M be a nonsimply connected topological space and let K be a base for the topologyof M as in the statement of Corollary 3.12. Then to any nontrivial group homomorphismin H(π1(M, x0), G) there corresponds a 1–cocycle of Z1(K, G) which is not a 1–coboundary.

4. Connections 1–cochains

This section is entirely devoted to studying the connection 1–cochains of a poset K andrelated notions like curvature, holonomy group and central connection 1–cochains. Theinflationary structure of Σ∗(K) enters the theory at this point. We shall show how con-nection 1–cochains and 1–cocycles are related, thus allowing one to interpret a 1–cocycleas a principal bundle and a connection 1-cochain as a connection on this principal bundle.We shall prove the existence of nonflat connection 1–cochains, a “poset” version of theAmbrose-Singer Theorem, and that to any flat connection 1-cochain with values in G,there corresponds a homomorphism from the fundamental group of the poset into G. Allthe definitions admit an obvious generalization to symmetric simplicial sets having aninflationary structure. However, the main result, the relation between connection andcocycles, holds only in a particular case, as we shall see.


4.1. Connections and curvature. We now give the definition of a connection 1–cochain of a poset with values in a group. To this end, recall the definition of the setΣinf

n (K) of inflating n–simplices (see Section 2).

4.2. Definition. A 1–cochain u of C1(K, G) is said to be a connection 1–cochain,or, simply, a connection, if it satisfies the following properties:

(i) u(b) = u(b)−1 for any b ∈ Σ1(K);

(ii) u(∂0c) u(∂2c) = u(∂1c), for any c ∈ Σinf2 (K).

We denote the set of connection with values in G by U1(K, G).

This definition of a connection is related to the notion of the link operator in a latticegauge theory ([6]) and to the notion of a generalized connection in loop quantum gravity([1, 13]). Both the link operator and the generalized connection can be seen as a mappingA which associates an element A(e) of a group G to any oriented edge e of a graph α,and enjoys the following properties

A(e) = A(e)−1, A(e2 ∗ e1) = A(e2) A(e1), (40)

where, e is the reverse of the edge e; e2 ∗ e1 is the composition of the edges e1, e2 obtainedby composing the end of e1 with the beginning of e2. Now, observe that to any poset Kthere corresponds an oriented graph α(K) whose set of vertices is Σ0(K), and whose setof edges is Σ1(K). Then, by property (i) of the above definition and property (29), anyconnection u ∈ U1(K, G) defines a mapping from the edges of α(K) to G satisfying (40).The new feature of our definition of connection, is to require property (ii) in Definition4.2, thus involving the poset structure. The motivation for this property will become clearin the next section: thanks to this property any connection u can be seen as a connectionon the principal bundle described by a 1-cocycle (see Theorem 4.12).

Let us now observe that any 1–cocycle is a connection. Furthermore, if u is a con-nection then so is any equivalent 1–cochain (the proof is similar to the proof of thecorresponding property for 1–cocycles, see Section 3.2).

4.3. Lemma. Let γ : G1 → G be a group homomorphism. Given v ∈ C1(K, G1) considerγ ◦v ∈ C1(K, G). Then: if v is a connection then γ ◦v is a connection; the converse holdsif γ is injective.

Proof. Clearly, if v is a connection so is γ ◦ v. Conversely, assume that γ is injective andthat γ ◦ v is a connection. If c ∈ Σinf

2 (K), then

γ(v(∂0c) v(∂2c)

)= γ ◦ v(∂0c) γ ◦ v(∂2c) = γ ◦ v(∂1c) = γ

(v(∂1c)

),

hence v(∂0c) v(∂2c) = v(∂1c), since γ is injective. Furthermore, for any 1–simplex b wehave

γ(v(b)) = γ ◦ v(b) = (γ ◦ v(b))−1 = γ(v(b)−1).

So, as γ is injective, we have v(b) = v(b)−1, and this entails that v is a connection.


4.4. Lemma. Given u ∈ U1(K, G), then u(σ0a) = e for any a ∈ Σ0(K).

Proof. Note that c ≡ σ1σ0a is an inflating 2–simplex with ∂ic = σ0a for i = 0, 1, 2. ByDefinition 4.2(ii) we have u(σ0a) u(σ0a) = u(σ0a) which is equivalent to u(σ0a) = e.

We call the full subcategory of C1(K, G) whose set of objects is U1(K, G) the category ofconnection 1–cochains with values in G. It will be denoted by the same symbol U1(K, G)as used to denote the corresponding set of objects. Note that Z1(K, G) is a full subcategoryof U1(K, G). Furthermore, if γ : G1 → G is a group homomorphism, by Lemma 4.3, therestriction of the functor γ◦ to U1(K, G1) defines a functor from U1(K, G1) into U1(K, G).

As observed, any 1–cocycle is a connection. The converse does not hold, in general,and the obstruction is a 2–coboundary.

4.5. Definition. The curvature of a connection u ∈ U1(K, G) is the 2–coboundaryWu ≡ du ∈ B2(K, G). Explicitly, by using relation (22) we have

(Wu)1(b) =(e, ad(u(b))

), b ∈ Σ1(K),

(Wu)2(c) =(wu(c), ad(u(∂1c)

), c ∈ Σ2(K),

where wu : Σ2(K) → G defined as

wu(c) ≡ u(∂0c) u(∂2c) u(∂1c)−1, c ∈ Σ2(K).

A connection u ∈ U1(K, G) is said to be flat whenever its curvature is trivial i.e. Wu ∈(2G)1 or, equivalently, if wu(c) = e for any 2–simplex c.

We now draw some consequences of our definition of the curvature of a connection andpoint out the relations of this notion to the corresponding one in the theory of principalbundles.

First, note that a connection u is flat if, and only if, u is a 1–cocycle. Then, as animmediate consequence of Proposition 3.11, we have a poset version of a classical resultof the theory of principal bundles [11, 7].

4.6. Corollary. There is, up to equivalence, a 1-1 correspondence between flat connec-tions of K with values in G and group homomorphisms from π1(K) into G.

The existence of nonflat connections will be shown in Section 4.22 where exampleswill be given.

Secondly, in a principal bundle the curvature form is the covariant exterior derivativeof a connection form, namely the 2–form with values in the Lie algebra of the group,obtained by taking the exterior derivative of the connection form and evaluating this onthe horizontal components of pairs of vectors of the tangent space (see [11]). Although,no differential structure is present in our approach, Wu encodes this type of information.In fact, given a connection u, if we interpret u(p) as the horizontal lift of a path p, thenthe equation

wu(c) u(∂1c) = u(∂0c ∗ ∂2c) wu(c), c ∈ Σ2(K), (41)


may be understood as saying that wu(c) intertwines the horizontal lift of the path ∂1c andthat of the path ∂0c ∗ ∂2c.

Thirdly, the structure equation of the curvature form (see [11]) says that the curvatureequals the exterior derivative of the connection form plus the commutator of the connec-tion form. Notice that the second component (Wu)2 of the curvature can be rewrittenas

(Wu)2(c) =(wu(c), ad(wu(c))

−1)×

(e, ad(u(∂0c)u(∂2c))

), c ∈ Σ2(K), (42)

where × is the composition (11) of the 2–category 2G. This equation represents, in ourformalism, the structure equation of the curvature with

(wu(c), ad(wu(c)

−1))

in placeof the exterior derivative, and

(ι, ad(u(∂0c)u(∂2c))

)in place of the commutator of the

connection form.

Fourthly, as a consequence of Lemma 3.3 we have that Wu is a 2–cocycle. The 2–cocycle identity, dWu ∈ (3G)2, or, equivalently,

wu(∂0d) wu(∂2d) = ad(u(∂01d))(wu(∂3d)

)wu(∂1d), d ∈ Σ3(K), (43)

corresponds to the Bianchi identity in our framework.

We conclude with the following result.

4.7. Lemma. For any connection u the following assertions hold:

(a) wu(τ0c) = wu(c)−1 for any 2–simplex c;

(b) wu(c) = e if c is either a degenerate or an inflating 2-simplex.

Proof. (a) follows directly from the definition of τ0c, see Section 2. (b) If c is an inflating2–simplex, then wu(c) = e because of Definition 4.2(ii). Given a 1–simplex b, then

wu(σ0(b)) = u(∂0σ0(b)) u(∂2σ0(b)) u(∂1σ0(b))−1 = u(b) u(σ0(∂1b)) u(b)−1 = e,

because u(σ0(∂1b)) = e (Lemma 4.4). Analogously we have that wu(σ1(b)) = e.

In words, statement (b) asserts that the curvature of a connection is trivial whenrestricted to inflating simplices.

4.8. Remark. It is worth pointing out some analogies between the theory of connections,as presented in this paper, and that developed in synthetic geometry by A. Kock [12],and in algebraic geometry by L. Breen and W. Messing [4]. The contact point with ourapproach resides in the fact that both of the other approaches make use of a combinatorialnotion of differential forms taking values in a group G. So in both cases connectionsturn out to be combinatorial 1–forms. Concerning the curvature, the definition of Wu

is formally the same as the definition of curving data given in [4], since this is the 2–coboundary of a connection, taking values in a 2–category associated with G. Whereas,in [12] the curvature is the 2–coboundary of a connection, taking values in G, and isformally the same as wu. A significant difference from these other two approaches is thatin our case wu is not invariant under oriented equivalence of 2-simplices (examples ofconnections having this feature will be studied in Section 4.19).


4.9. The cocycle induced by a connection. We analyze the relation between con-nections and 1–cocycles more deeply. The main result is that to any connection therecorresponds a unique 1–cocycle. This, on the one hand, confirms the interpretation of1–cocycles as principal bundles. On the other hand this result will allow us to constructexamples of nonflat connections in Section 4.22.

To begin with, we need a preliminary definition. We call the collection {Va, a ∈Σ0(K)} of open subsets of K defined by

Va ≡ {O ∈ K | a ≤ O}, a ∈ Σ0(K), (44)

the minimal open covering of K. It is an open covering because any element of K belongsto some Va, and because any Va is an open subset of K for the Alexandroff topology (seeSubsection 3.5). It is minimal because if U is an open covering of K, then any Va iscontained in some element of U .

We now turn to study the relation between connections and 1–cocycles.

4.10. Lemma. For any u ∈ U1(K, G), there exists a unique family {za0,a1} of functionsza0,a1 : Va0 ∩ Va1 → G satisfying the following two properties:

(i) z∂0b,∂1b(∂0b) = u(b) if b ∈ Σinf1 (K);

(ii) za2,a1(a) za1,a0(a) = za2,a0(a) if a ∈ Va0 ∩ Va1 ∩ Va2.

Moreover, (i) and (ii) imply that

(iii) za0,a1(a) = za0,a1(a′), if a ≤ a′ and a ∈ Va0 ∩ Va1.

Proof. Recall that (a0, a1) denotes the inflating 1–simplex b such that ∂0b = a0 and∂1b = a1 (see Subsection 2.4). Given u ∈ U1(K, G), define

za0,a1(a) ≡ u(a, a0)−1 u(a, a1), a ∈ Va0 ∩ Va1 , (45)

(i) Let b ∈ Σinf1 (K). By Lemma 4.4 we have

z∂0b,∂1b(∂0b) = u(∂0b, ∂0b)−1 u(∂0b, ∂1b) = u(σ0∂0b)

−1 u(b) = u(b).

(ii) Let a ∈ Va0 ∩ Va1 ∩ Va2 . Then

za2,a1(a) za1,a0(a) = u(a, a2)−1 u(a, a1) u(a, a1)

−1 u(a, a0)

= u(a, a2)−1 u(a, a0)

= za2,a0(a).

Before proving uniqueness, we observe that by (ii), za0,a1(a) = za1,a0(a)−1 for any a ∈Va0 ∩ Va1 . Now, let {z′a0,a1

} be another family of functions satisfying the above relations.


By (i), {z′a0,a1} and {za0,a1} must agree on inflating 1–simplices. Using this observation

and (ii), we have

z′a0,a1(a) = z′a0,a(a) z′a,a1

(a) = z′a,a0(a)−1 z′a,a1

(a) = za,a0(a)−1 za,a1(a) = za0,a1(a)

for any a ∈ Va0 ∩ Va1 . Finally, we prove that (i) and (ii) imply (iii). Let a ≤ a′ anda ∈ Va0 ∩ Va1 . By (i) and (ii) we have

za0,a1(a) = za0,a(a) za,a1(a) = u(a, a0)−1 u(a, a1).

As observed after Definition 2.6, there are c, c′ ∈ Σinf2 (K) such that ∂2c = (a, a1), ∂0c =

(a′, a), ∂1c = (a′, a1), and ∂2c′ = (a, a0), ∂0c

′ = (a′, a), ∂1c′ = (a′, a0). Then

za0,a1(a) =(u(∂0c

′)−1 u(∂1c′))−1

u(∂0c)−1 u(∂1c) = u(a′, a0) u(a′, a1) = za0,a1(a

′),

and this completes the proof.

4.11. Remark. Lemma 4.10 is the contact point with Cech cohomology. In fact, one canunderstand the family of functions {za0,a1}, defined by (45), as a 1–cocycle of a poset, inthe sense of the Cech cohomology, with respect to the minimal covering of the poset. Ina forthcoming paper [21], we shall see that such functions are nothing but the transitionfunctions of a principal bundle over a poset. We shall also see that such bundles can bemapped in to locally constant bundles over M when the poset K is a base for the topologyof a topological space M .

We now prove the main result of this section.

4.12. Theorem. For any u ∈ U1(K, G), there exists a unique 1–cocycle z ∈ Z1(K, G)such that

u(b) = z(b), b ∈ Σinf1 (K).

Proof. Consider the family of function {za0,a1} associated with u by Lemma 4.10. Notethat for any 1–simplex b, since ∂0b, ∂1b ≤ |b|, we have that |b| ∈ V∂0b ∩V∂1b. Hence we candefine

z(b) ≡ z∂0b,∂1b(|b|), b ∈ Σ1(K). (46)

Given a 2–simplex c and using properties (ii) and (iii) of {za0,a1}, we have

z(∂0c) z(∂2c) = z∂00c,∂10c(|∂0c|) z∂02c,∂12c(|∂2c|) = z∂00c,∂10c(|c|) z∂02c,∂12c(|c|)= z∂01c,∂10c(|c|) z∂10c,∂11c(|c|) = z∂01c,∂11c(|c|) = z∂01c,∂11c(|∂1c|)= z(∂1c).

If b ∈ Σinf1 (K), by property (i) of {za0,a1} we have z(b) = z∂0b,∂1b(∂0b) = u(b). Uniqueness

is obvious.


On the basis of Theorem 4.12 we can introduce the following definition.

4.13. Definition. A connection u ∈ U1(K,G) is said to induce the 1–cocycle z ∈Z1(K, G) whenever

u(b) = z(b), b ∈ Σinf1 (K) .

We denote the set of connections of U1(K, G) inducing the 1–cocycle z by U1(K, z).

The geometrical meaning of U1(K, z) is the following: just as a 1–cocycle z standsfor a principal bundle over K so the set of connections U1(K, z) stands for the set ofconnections on that principal bundle. Theorem 4.12 says that the set of connections withvalues in G is partitioned as

U1(K, G) = ∪{U1(K, z) | z ∈ Z1(K, G)

}(47)

where the symbol ∪ means disjoint union.

4.14. Lemma. Given z1, z ∈ Z1(K, G), let u1 ∈ U1(K, z1) and u ∈ U1(K, z). Then(u1, u) ⊆ (z1, z). In particular if u1

∼= u, then z1∼= z.

Proof. Given a 1–simplex b, consider the inflating 1–simplices (|b|, ∂1b) and (|b|, ∂0b). Byequations (45) and (46), we have

z(b) = u(|b|, ∂0b)−1 u(|b|, ∂1b).

The same holds for z1 and u1. Given f ∈ (u1, u), we have

f∂0b z1(b) = f∂0b u1(|b|, ∂0b) u1(|b|, ∂1b)

= u1(|b|, ∂0b) f|b| u1(|b|, ∂1b) = u(|b|, ∂0b) u(|b|, ∂1b) f∂1b

= z(b) f∂1b,

where (|b|, ∂0b) is the reverse of (|b|, ∂0b). Hence f ∈ (z1, z).

Now, given a 1-cocycle z ∈ Z1(K, G), we call the category of connections inducing z,the full subcategory of U1(K, G) whose objects belong to U1(K, z). As it is customary inthis paper, we denote this category by the same symbol U1(K, z) as used to denote thecorresponding set of objects.

4.15. Lemma. Let z ∈ Z1(K, G1) and let γ : G1 → G be an injective group homomor-phism. Then, the functor γ◦ : U1(K, z) → U1(K, γ ◦ z) is injective and faithful.

Proof. Given u ∈ U1(K, z), it is easy to see that γ ◦ u ∈ U1(K, γ ◦ z). Clearly, as γ isinjective, the functor γ◦ is injective and faithful.


We note the following simple result.

4.16. Lemma. If z1∼= z, then the categories U1(K, z1) and U1(K, z) are equivalent.

Assume that K is simply connected. In this case any 1-cocycle is a 1–coboundary (seeSection 3.5). Then the category U1(K, z) is equivalent to U1(K, ı) for any z ∈ Z1(K, G).

The case of inflationary symmetric simplicial sets. The notions of a connectionand curvature admit a straightforward generalization to a symmetric simplicial set Σ∗which is pathwise connected and has an inflationary structure Σ′

∗. We now show thatboth Lemma 4.10 and Theorem 4.12 admit a generalization to this kind of simplicial sets.The latter, however, with an additional assumption.

Denote the set 1–cocycles of Σ∗ with values in G by Z1(Σ∗, G), and the set of connec-tions of Σ∗ with values in G by U1(Σ∗, G). Moreover, let Va = {a ∈ Σ0 | a ≤ a}, where ≤is the order relation on Σ0 induced by the inflationary structure of Σ∗ (Lemma 2.7).

4.17. Lemma. For any u ∈ U1(Σ∗, G), there exists a unique family {za0,a1} of functionsza0,a1 : Va0 ∩ Va1 → G satisfying the properties (i)–(iii) of Lemma 4.10.

Proof. Given a ∈ Va0 ∩ Va1 , define

za0,a1(a) ≡ u(a, a0)−1 u(a, a1), a ∈ Va0 ∩ Va1 .

The proof of the properties (i) and (ii), uniqueness, and property (iii) follows like theproof of Lemma 4.10.

It is worth observing that in Theorem 4.12 we have used the support |b| of a 1–simplexb to define the cocycle induced by a connection. Actually, |b| is a preferred element ofV∂0b ∩ V∂1b, because of the property that |b| ≤ |c| if b = ∂ic for some c ∈ Σ2(K) andfor some i. Such an element is, in general, missing in an arbitrary inflationary simplicialset Σ∗. Hence in order to define the 1–cocycle induced by a connection we shall need anadditional assumption on Σ∗.

4.18. Theorem. Let u ∈ U1(Σ∗, G). Assume that for any 1–simplices b the set V∂0b∩V∂1b

is pathwise connected. Then, there exists a unique 1–cocycle z ∈ Z1(Σ∗, G) such thatz(b) = u(b) for any b ∈ Σ′

1.

Proof. Given a 1–simplex b, observe that by property (v) of inflating simplices, the setV∂0b ∩ V∂1b is never empty. Define,

z(b) ≡ z∂0b,∂1b(a), a ∈ V∂0b ∩ V∂1b.

We show that this definition does not depend on the choice of a. Let a ∈ V∂0b ∩ V∂1b.

Assume that there is b ∈ Σ1 such that ∂1b = a and ∂0b = a. By property (v) of inflatingsimplices, there is a 0–simplex a′ such that a, a ≤ a′. By property (iii) of {za0,a1} we have

z∂0b,∂1b(a) = z∂0b,∂1b(a′) = z∂0b,∂1b(a).


As V∂0b ∩ V∂1b is pathwise connected the proof follows. Given a 2–simplex c, property (v)of inflating n–simplices implies that there is a 0–simplex a greater than ∂01c, ∂02c, ∂12c.Using properties (ii) and (iii) of {za0,a1} we have

z(∂0c) z(∂2c) = z∂00c,∂10c(a) z∂02c,∂12c(a)

= z∂00c,∂02c(a) z∂02c,∂12c(a) = z∂00c,∂12c(a)

= z∂01c,∂11c(a) = z(∂1c).

Hence z is a 1–cocycle. The rest of the proof follows straightforwardly.

4.19. Central connections. We now briefly study the family of central connections,whose main feature, as we shall show below, is that any such connection can be uniquelydecomposed as the product of the induced cocycle by a suitable connection taking valuesin the centre of the group.

4.20. Definition. A connection u ∈ U1(K, G) is said to be a central connectionwhenever the component wu of the curvature Wu takes values in the centre Z(G). Wedenote the set of central connections by U1

Z(K, G).

Let us start to analyze the properties of central connections. Clearly 1–cocycles arecentral connections. However, the main property that can be directly deduced from theabove definition is that the component wu of the curvature Wu of a central connection uis invariant under oriented equivalence of 2–simplices. In fact by the definition of wu, itis easily seen that

wu(c) = wu(c1) = wu(c2)−1, c1 ∈ [c], c2 ∈ [c], (48)

for any 2–simplex c, where [c] and [c] are, respectively, the classes of 2–simplices havingthe same and the reversed orientation of c.

4.21. Proposition. A connection u of U1(K, G) is central if, and only if, it can beuniquely decomposed as

u(b) = zu(b) χu(b), b ∈ Σ1(K),

where zu ∈ Z1(K, G), and χu ∈ U1(K, ı) with values in Z(G).

Proof. (⇐) Assume that a connection u admits a decomposition as in the statement.Since χu takes values in the centre, so does wu. Furthermore, since χu ∈ U1(K, ı) thenχu(b) = e for any inflating 1–simplex b. This entails that zu is nothing but the 1–cocycleinduced by u. This is enough for uniqueness. (⇒) Assume that u is central. For any1–simplex b let cb denote the 2–simplex defined as

∂1cb ≡ b, ∂0cb ≡ (|b|, ∂0b), ∂2cb ≡ (|b|, ∂1b), |cb| ≡ |b|.

Define zu(b) ≡ u(b) wu(cb), and observe that, as wu takes values in Z(G), we have

zu(b) = u(b) wu(cb) = u(∂1cb) wu(cb) = u(∂0cb) u(∂2cb) = u(|b|, ∂0b)−1 u(|b|, ∂1b).


Hence zu is the 1–cocycle induced by u. Now, define

χu(b) ≡ wu(cb), b ∈ Σ1(K).

Since χu(b) = u(b) zu(b)−1, one can easily deduce that χu ∈ U1(K, ı), and this completes

the proof.

As a consequence of this result the set U1Z(K, z) of central connections inducing the

1–cocycle z, has a the structure of an Abelian group. In fact, given u, u1 ∈ U1Z(K, z),

defineu ?z u1(b) ≡ u(b) z(b)−1 u1(b), b ∈ Σ1(K). (49)

By Proposition 4.21, we have u ?z u1(b) = z(b) χu(b) χu1(b) for any 1–simplex b. Thisentails that

u ?z u1 = u1 ?z u and u ?z u1 ∈ U1Z(K, z).

By this relations, it is easily seen that U1Z(K, z) with ?z is an Abelian group whose identity

is z, and such that the inverse of a connection u is the connection defined as z(b) χu(b)−1

for any 1–simplex b.

Finally, in Section 4.1 we pointed out the analogy between equation (42) and thestructure equation of the curvature of a connection in a principal bundle. This analogyis stronger for a central connection u since we have

(Wu)2(c) =(wu(c), ι

)×

(e, ad(u(∂0c)u(∂2c))

)=

(e, ad(u(∂0c)u(∂2c))

)×

(wu(c), ι

),

(50)

for any 2–simplex c. Hence, as for principal bundles, equation (42) for a central connectionis symmetric with respect to the interchange of the two factors.

4.22. Existence of nonflat connections. We investigate the existence of nonflatconnections. As a first step, we show that there is a very particular class of posets notadmitting nonflat connections.

Recall that a poset K is said to be totally ordered whenever for any pair O,O1 ∈ Keither O ≤ O1 or O1 ≤ O. Clearly, a totally ordered poset is directed and, consequently,pathwise connected (it is also simply connected, see Section 2).

4.23. Corollary. If K is totally ordered, any connection is flat.

Proof. If K is totally ordered and b is any 1–simplex either b or b is an inflating 1–simplex. Hence, by Theorem 4.12 any connection is equal to the induced 1–cocycle.

Another obvious situation where nonflat connections do not exist is when the group ofcoefficients G is trivial, i.e. G = e. Two observations on these results are in order. First,Corollary 4.23 cannot be directly deduced from the definition of a connection. Secondly,as explained earlier, these two situations never arise in the applications we have in mind.

Now, our purpose is to show that, except when the poset is totally ordered or thegroup of coefficients is trivial, nonflat connections always exist. Let us start with thefollowing


4.24. Lemma. Let v ∈ C1(K, G) be such that v(b) = e = v(b) for any inflating 1–simplexb. Then, for any 1–cocycle z ∈ Z1(K, G) the 1–cochain v(z) defined as

v(z)(b) ≡ v(b)−1 z(b) v(b), b ∈ Σ1(K), (51)

is a connection inducing z.

Proof. By the definition of v for any inflating 1–simplex b we have that

v(z)(b) = v(b)−1 z(b) v(b) = e z(b) e = z(b) .

This, in particular, entails that v(z) satisfies property (ii) of the definition of connections.For any 1–simplex b we have

v(z)(b) = v(b)−1 z(b) v(b) = v(b)−1 z(b)−1 v(b) =(v(b)−1 z(b) v(b)

)−1= v(z)(b)−1.

Hence v(z) ∈ U1(K, z).

It is very easy to prove the existence of 1–cochains satisfying the properties of thestatement. For instance, given a 1–simplex b, define

v(b) ≡{

e b or b ∈ Σinf1 (K)

g(b) otherwise ,(52)

where g(b) is some element of the group G. So v is a 1–cochain satisfying the relationv(b) = e = v(b) for any inflating 1–simplex b.

Now, assume that K is a pathwise connected but not totally ordered poset. Let G bea nontrival group. Let v ∈ C1(K, G) be defined by (52), and let z ∈ Z1(K, G). Considerthe connection v(z) ∈ U1(K, z). We want to find conditions on v implying that v(z) isnot flat.

As v(z) ∈ U1(K, z), Theorem 4.12 says that if v(z) is flat then v(z) = z. Hence, v(z)is not flat if, and only if, it differs from z on a 1–simplex b such that both b and b are notinflating. Then

v(z)(b) 6= z(b) ⇐⇒ v(b)−1 z(b) v(b) 6= z(b)

⇐⇒ z(b) v(b) 6= v(b) z(b) ⇐⇒ z(b) g(b) 6= g(b) z(b)

So, for instance, if we take

g(b) = z(b)−1 and g(b) = g z(b)−1 with g 6= e,

then v(z) is not flat. Note that the above choice is always possible because G is nontrivialby assumption. In conclusion we have shown the following

4.25. Theorem. Let K be a pathwise connected but not totally ordered poset. Let G be anontrivial group. Then for any 1–cocycle z ∈ Z1(K, G) there are connections in U1(K, z)which are not flat.

Concerning central connections, in the case that K and G satisfy the hypotheses ofthe statement of Theorem 4.25, and the centre of the group G is nontrivial, then by usingthe above reasoning it is very easy to prove the existence of nonflat central connections.


4.26. Holonomy and reduction of connections. Keeping close to the theory offibre bundles, we introduce the notion of holonomy group of a connection and link to itthe property of reducibility of a connection.

Consider a connection u of U1(K, G). Fix a base 0–simplex a and define

Hu(a) ≡{u(p) ∈ G | p : a → a

}, (53)

recalling that p : a → a denotes i a loop of K with endpoint a. By the defining propertiesof connections it is very easy to see that Hu(a) is a subgroup of G. Furthermore let

H0u(a) ≡

{u(p) ∈ G | p : a → a, p ∼ σ0a

}, (54)

where p ∼ σ0a means that p is homotopic to the degenerate 1–simplex σ0a. In this case,too, it is easy to see that H0

u(a) is a subgroup of G. Moreover, since p ∗ q ∗ p ∼ σ0awhenever q, p ∈ (a, a) and q ∼ σ0a, H0

u(a) is a normal subgroup of Hu(a). Hu(a) andH0

u(a) are called respectively the holonomy and the restricted holonomy group of u basedon a.

As K is pathwise connected, we have the following

4.27. Lemma. Given u ∈ U1(K, G), let γ : G1 → G be an injective homomorphism. Thefollowing assertions hold.

(a) Hu(a) and Hu(a1) are conjugate subgroups of G for any a, a1 ∈ Σ0(K).

(b) Given u1 ∈ U1(K, G1). If γ ◦u1 is equivalent to u, then the holonomy groups Hu1(a)and Hu(a) are isomorphic.

The same assertions hold for the restricted holonomy groups.

Proof. (a) Let p be a path from a to a1. For any g ∈ Hu(a), there is a loop q : a → a suchthat g = u(q). Observe that p∗q ∗p : a1 → a1, hence u(p) g u(p)−1 = u(p∗q ∗p) ∈ Hu(a1).By the symmetry of the reasoning, Hu(a) 3 g → u(p) g u(p)−1 ∈ Hu(a1) is a groupisomorphism. (b) Let u1 ∈ U1(K, G1) and let f ∈ (γ ◦u1, u). Since for any loop p : a → a,fa γ ◦u1(p) = u(p) fa, the map Hu1(a) 3 g → fa γ(g) f−1

a ∈ Hu(a) is a group isomorphism.

Now, according to the definition given in Section 3.5, a connection u ∈ U1(K, G) isreducible if there is a proper subgroup G1 of G and a connection u1 ∈ U1(K, G1) suchthat ◦ u1

∼= u, where : G1 → G is the inclusion mapping. The next result, an analogueof the Ambrose-Singer theorem [11] for connections of a poset, shows that u is reduciblewhenever its holonomy group is a proper subgroup of G.

4.28. Theorem. Given z ∈ Z1(K, G), let u ∈ U1(K, z). Given a0 ∈ Σ0(K), let :Hu(a0) → G be the inclusion mapping. Then,

(a) there exists z1 ∈ Z1(K, Hu(a0)) such that ◦ z1∼= z,

(b) there exists u1 ∈ U1(K, z1) such that ◦ u1∼= u.


Proof. For any 0–simplex a, let pa be a path from a0 to a. Then define

u1(b) ≡ u(p∂0b ∗ b ∗ p∂1b), b ∈ Σ1(K).

Note that u1(b) ∈ Hu(a0) for any 1–simplex b because p∂0b ∗ b ∗ p∂1b : a0 → a0. Secondly,for any 1–simplex b we have

u1(b) = u(p∂1b ∗ b ∗ p∂0b) = u(p∂0b ∗ b ∗ p∂1b) = u1(b)−1.

Thirdly, let c ∈ Σinf2 (K). Then

u1(∂0c) u1(∂2c) = u(p∂00c ∗ ∂0c ∗ p∂10c) u(p∂02c ∗ ∂2c ∗ p∂12c)

= u(p∂00c) u(∂0c) u(p∂10c) u(p∂02c) u(∂2c) u(p∂12c)

= u(p∂01c) u(∂0c) u(∂2c) u(p∂11c)

= u(p∂01c) u(∂1c) u(p∂11c)

= u1(∂1c).

Therefore we have that u1 ∈ U1(K, Hu(a0)). Now, for any 0–simplex a let fa ≡ u(pa).Then for any 1–simplex b we have

f∂0b u1(b) = u(p∂0b) u(p∂0b ∗ b ∗ p∂1b) = u(p∂0b) u(p∂0b) u(b) u(p∂1b) = u(b) f∂1b,

namely f ∈ ( ◦ u1, u). Thus ◦ u1∼= u. Finally, let z1 ∈ Z1(K, Hu(a0)) be the 1–cocycle

induced by u1. Then ◦ z1∼= z because of Lemma 4.14. This completes the proof

5. Gauge transformations

In the previous sections we have given several results to support the interpretation of1–cocycles of a poset as principal bundles over the poset. As the final issue of the presentpaper, we now introduce what we mean by the group of gauge transformations of a 1–cocycle.

Given a 1–cocycle z of Z1(K, G), define

G(z) ≡ (z, z). (55)

An element of G(z) will be denoted by g. The composition law between morphisms of1–cochains endows G(z) with a structure of a group. The identity e of this group is givenby ea = e for any 0–simplex a. The inverse g−1 of an element g ∈ G(z) is obtained bycomposing g with the inverse of G. We call G(z) the group of gauge transformations of z.

5.1. Lemma. If z ∈ B1(K, G), then G(z) ∼= G.

Proof. Observe that, since K is connected, G(ı) is the set of constant functions fromΣ0(K) to G and hence is isomorphic to G. As z is a 1–coboundary, it is equivalent to thetrivial 1–cocycle ı, i.e. there exists an f ∈ (z, ı). The mapping G(ı) 3 g 7→ f−1 g f ∈ G(z)is a group isomorphism.


As a consequence of this lemma and Proposition 3.11, if the poset is simply connectedthen G(z) ∼= G for any 1–cocycle z. This is also the case when G is Abelian.

5.2. Lemma. If G is Abelian, then G(z) ∼= G for any z ∈ Z1(K, G).

Proof. For any g ∈ G(z) and for any 1–simplex b we have

g∂1b z(b) = z(b) g∂0b = g∂0b z(b) .

Hence g∂1b = g∂0b for any 1–simplex b. Since K is pathwise connected, ga = g for any0–simplex a.

Thus, for Abelian groups, the action of the group of gauge transformations is alwaysglobal, that is independent of the 0–simplex.

Given a 1–cocycle z ∈ Z1(K, G) consider the group G(z) of gauge transformations ofz. For any u ∈ U1(K, z) and g ∈ G(z), define

αg(u)(b) ≡ g∂0b u(b) g−1∂1b, b ∈ Σ1(K). (56)

We have the following

5.3. Proposition. Given z ∈ Z1(K, G), the following assertions hold:

(a) given g ∈ G(z), then αg(u) ∈ U1(K, z) for any u ∈ U1(K, z);

(b) The mapping

α : G(z)× U1(K, z) 3 (g, u) −→ αg(u) ∈ U1(K, z) (57)

defines a left action, not free, of G(z) on U1(K, z).

Proof. (a) Clearly αg(u)(b) = αg(u)(b)−1 for any 1–simplex b. Moreover, if b ∈ Σinf1 (K),

then αg(u)(b) = g∂0b u(b) g−1∂1b = g∂0b z(b) g−1

∂1b = z(b). This entails that αg(u) satisfiesproperty (ii) of the definition of connections. Hence αg(u) ∈ U1(K, z). (b) Clearly, α is aleft action that is not free, because z ∈ U1(K, z), hence αg(z) = z for any g ∈ G(z).

6. Conclusions and outlook

We have developed a theory of bundles over posets from a cohomological standpoint, theanalogue of describing the usual principal bundles in terms of their transition functions. Ina sequel, we will introduce principal bundles over posets and their mappings directly andfurther develop such concepts as connection, curvature, holonomy, transition function,gauge group and gauge transformation. Although all these concepts are familiar fromthe usual theory of principal bundles, at this point it is worth stressing some of thedifferences from that theory. As we shall see in the sequel, the definition of principalbundle involves bijections between different fibres satisfying a 1–cocycle identity. Animportant role is played by the simplicial set of inflationary simplices. All principal


bundles can be trivialized on the minimal covering. Finally, it should be stressed thatthe goal of these investigations is to develop gauge theories in the framework of algebraicquantum field theory. Our principal fibre bundles and the associated vector bundles areenvisaged as stepping stones to the algebra of observables.

A. Some results on n–categories

In this appendix we explain the definition of the category I(C), and how to derive thecategories nG and the form of a 3–cochain. References for this appendix are [16, 23, 2, 19].

The category I(C). Consider an n–category C and let � be a composition law of C.Given an arrow t, the left and the right �–units of t are the arrows l�(t) r�(t) of C satisfyingthe relations

l�(t) � t = t = t � r�(t).

Given another composition law ×, with × ≺ �, the mappings r� : C 3 t → r�(t) ∈ C andl� : C 3 t → l�(t) ∈ C satisfy the following properties:

1. r�r× = r× = r×r� = r×l� and l× = l×r� = l×l�;

2. if r×(s) = l×(t), then

r�(s× t) = r�(s)× r�(t) and l�(s× t) = l�(s)× l�(t);

3. r�(s) = l�(t), r�(s1) = l�(t1) and r×(s � t) = l×(s1 � t1) entail that

(s � t)× (s1 � t1) = s× s1 � t× t1;

that is the interchange law,

see for instance [23] where the mapping r� and l� are called, respectively, source and targetof �.

From now on we assume that C is an n–category satisfying the following property:the arrows of C are invertible with respect to the greatest composition law. Recall that anarrow t is said to be �–invertible if there is an arrow t−1

� , called the �–inverse of t, suchthat t−1

� � t = r�(t), and t � t−1� = l�(t). Now, let � denote the greatest composition law of

C. Given an arrow t, define

R�(t) ≡ {m ∈ C | r�(m) = r�(t) = l�(m)}, (A.1)

and call the mapping βt : R�(t) → C, defined by

βt(m) ≡ t �m � t−1� , m ∈ R�(t), (A.2)

the inner �–automorphism associated with t. Since all the arrows of C are �–invertible,the set R�(t) is a group with respect to the composition law �. We call this group, the


domain of βt. Note that the inner �–automorphism βt determines t only up to elementsof the centre Z(R�(t)) of the group R�(t).

Consider now a pair of arrows s and t such that s × t is defined. By applying theexchange law we have

s× t � s−1� × t−1

� = (s � s−1� )× (t � t−1

� ) = l�(s)× l�(t) = l�(s× t)

that is (s × t)−1� = s−1

� × t−1� . So that the inner �–automorphism βs×t associated with

s × t can be written as βs×t(n) = s × t � n � s−1� × t−1

� for n ∈ R�(s × t). Now, for anypair of arrow t,s such that s× t is defined, we define

(βs × βt)(n) ≡ βs×t(n), n ∈ R�(s× t). (A.3)

We have seen that an inner �–automorphism is uniquely determined only up centralelements. Then, the above definition is well posed if we show that βs × βt dependsonly on βs and βt and not on the choice of the elements t, s which define these two �–automorphisms. To this end, we first observe that given z ∈ Z(R�(s)) and z′ ∈ Z(R�(t))then z × z′ is defined. This is easily seen by applying the relations between the sourceand target map, given at the beginning of this appendix, to the following identities

r×(s) = l×(t), r�(s) = r�(z) = l�(z), r�(t) = r�(z′) = l�(z

′).

We now have the following

A.1. Lemma. Let C be an n–category and let � be the greatest composition law of C.Assume that:

(i) the arrows of C are �–invertible;

(ii) for any composition law ×, with × ≺ �, we have

Z(R�(s))×Z(R�(t)) ⊆ Z(R�(s× t)).

for any pair of arrows t, s such that s× t is defined.

Then, for any pair of arrows s, t such that s× t is defined the composition βs × βt givenby (A.3), is well defined.

Proof. Let z ∈ Z(R�(s)), z′ ∈ Z(R�(t)), and let n ∈ R�(s× t). By property (ii) in thestatement and by applying the exchange law we have

(βs�z × βt�z′)(n) = β(s�z)×(t�z′)(n)

= (s � z)× (t � z′) � n �((s � z)× (t � z′)

)−1

�

= s× t � z × z′ � n �(s× t � z × z′

)−1

�

= s× t � z × z′ � n �(s× t � z × z′

)−1

�

= s× t � z × z′ � n �(z × z′

)−1

� �(s× t)−1

�

= s× t � n �(s× t

)−1

�

= βs×t(n) = (βs × βt)(n),


Completing the proof.

Now, let C be an n–category C satisfying the hypotheses of Lemma A.1, and let �denote the greatest composition law. Following [19], we define the category I(C) as theset

I(C) ≡ {(t, µ) ∈ C × Inn�(C) | t � t and βt � µ are defined}, (A.4)

where Inn�(C) is the set of inner �–automorphisms of C, with the following compositionlaws:

(i) (t, µ)× (t′, µ′) ≡ (t× t′, µ× µ′) if µ× µ′ is defined;(ii) (t, µ) � (t′, µ′) ≡ (t � µ(t′), µ � µ′) if µ � µ′ is defined;(iii) (t, µ) · (t′, µ′) ≡ (t � t′, µ′) if βt′ � µ′ = µ,

(A.5)

where × is any other composition law of C with × ≺ �. Lemma A.1 entails that thecomposition law × in I(C) is well defined. It turns out that I(C) is an (n + 1)–categorywith × ≺ � ≺ ·. The arrows of I(C) are invertible with respect the composition laws ·and �. However, in general, I(C) does not fulfill hypothesis (ii) of Lemma A.1.

The categories nG. Given a group G, consider the 1–category 1G whose arrows arethe elements of the group G and whose composition law is the group composition. Clearly1G satisfies the hypotheses of Lemma A.1, so we can define 2G ≡ I(1G). Since the arrowsof 1G are all composable, then any element of 2G is of the form (g, τ) with g ∈ G andτ ∈ Inn(G), where Inn(G) is the group of inner automorphisms of G. Furthermore, byA.5(ii) and A.5(iii), one can deduce the following composition laws

(i) (g, τ)× (g′, τ ′) = (gτ(g′), ττ ′),(ii) (g, τ) � (g′, τ ′) = (gg′, τ ′), if σg′τ ′ = τ.

(A.6)

That is, the composition laws × and � are obtained from the composition law of 1Gaccording to the definition (A.5)(ii) and (A.5)(iii) respectively. Notice that × is every-where defined; × ≺ �; the arrows of 2G are invertible with respect to any compositionlaw. Moreover, given an arrow (g, τ) of 2G, we have

l�(g, τ) = (e, τ), r�(g, τ) = (e, σgτ), (g, τ)−1� = (g−1, σgτ).

while

l×(g, τ) = (e, ι) = r×(g, τ), (g, τ)−1× = (τ−1(g−1), ι).

Hence one can see that 1–arrows are the elements of 2G of the form (e, τ), and that theonly 0–arrow is (e, ι), where ι is the identity automorphism.

Before studying the 3–category I(2G) we need some observations on the structureof the inner �–automorphisms of 2G. According to definition (A.1), the domain of aninner �–automorphism β(h,τ) of 2G, is the set of those arrows (m, γ) of 2G such that thecompositions

(m, γ) � (h, τ) and(h, τ) � (m, γ)


are defined. By (A.6)(ii), these compositions are defined whenever τ = γ and m ∈ Z(G).Hence, the domain of β(h,τ) is

R�(h, τ) = {(m, τ), m ∈ Z(G)}. (A.7)

Moreover, we have that

β(h,τ)(m, τ) = (m, σhτ), (m, τ) ∈ R�(h, τ). (A.8)

Note that R�(h, τ) is an Abelian group and that

R�(h, τ)× R�(h′, τ ′) = R�(hτ(h′), ττ ′). (A.9)

This equation implies that 2G satisfies the hypotheses of Lemma A.1, and we can definethe 3–category I(2G).

By (A.4), an arrow of I(2G) is a 3-tuple (g, α, β(h,τ)), where (g, α) ∈ 2G, and β(h,τ) isthe inner �–automorphism of 2G associated with (h, τ), such that the compositions

(g, α) � (g, α) and (g, α) � (h, τ).

are defined. By (A.6)(ii), these two compositions are defined if, and only if, σhτ = α andσgα = α. The latter, in particular, says that g is an element of the centre Z(G) of G.Hence

I(2G) = {(g, σhτ, β(h,τ)) | g ∈ Z(G), h ∈ G, τ ∈ Inn(G)}. (A.10)

The purpose now is to provide an explicit form to the abstract composition laws (A.5) ofI(2G). Let

(g, σhτ, β(h,τ)

),(g′, σh′τ ′, β(h′,τ ′)

)∈ I(2G). First, we consider the composition

law × as defined by (A.5)(i). Recalling definition (A.3), by (A.6) we have(g, σhτ, β(h,τ)

)×

(g′, σh′τ ′, β(h′,τ ′)

)=

(gg′, σhτσh′τ ′, β(hτ(h′),ττ ′)

). (A.11)

This composition is everywhere defined because the composition law × in 2G is. Considernow the composition law � as defined by (A.5)(ii). Note that, by (A.6)(ii), � is definedin I(2G) whenever β(h,τ) � β(h′,τ ′) is defined. This amounts to saying that (h, τ) � (h′, τ ′)is defined, which is equivalent to the condition σh′τ ′ = τ . Hence, one can check that(

g, σhτ, β(h,τ)

)�

(g′, σh′τ ′, β(h′,τ ′)

)=

(gg′, σhh′τ ′, β(hh′,τ ′)

), if σh′τ ′ = τ. (A.12)

Finally, consider the composition · as defined by (A.5)(iii). Again, by (A.6)(ii), · isdefined in I(2G) whenever

β(h,τ) = β(g′,σh′τ ′) � β(h′,τ ′) = β(g′,σh′τ ′)�(h′,τ ′) = β(g′h′,τ ′).

By (A.7) and (A.8), this relation is equivalent to τ = τ ′, and σhτ = σg′h′τ ′. The latter, inparticular, entails that σh = σh′ since g′ is an element of the centre of G. In conclusionwe have that(

g, σhτ, β(h,τ)

)·(g′, σh′τ ′, β(h′,τ ′)

)=

(gg′, σh′τ ′, β(h′,τ ′)

), if σh = σh′ , τ = τ ′. (A.13)

Now, it is very easy to see that 2–arrows are those elements of I(2G) of the form(e, σhτ, β(h,τ)); 1–arrows are of the form (e, τ, β(e,τ)); the only 0–arrow is (e, ι, β(e,ι)).

The 3–category 3G used in Section 3.2 as a set of coefficients for the cohomology ofposets is different from I(2G). We now show that these two 3–categories are isomorphic.


A.2. Lemma. The mapping F : I(2G) → 3G defined by

F(g, σhτ, β(h,τ)

)≡ (g, τ, σh),

for any(g, σhτ, β(h,τ)

)∈ I(2G), is an isomorphism of 3–categories.

Proof. It is clear that the mapping F is injective and surjective. We check that Fpreserves the composition laws. To this end, fix a pair

(g, σhτ, β(h,τ)

),(g′, σh′τ ′, β(h′,τ ′)

)of arrows of I(2G).

Concerning the composition law ·, by (13) we have


)· F

(g′, σh′τ ′, β(h′,τ ′)

)= (g, τ, σh) · (g′, τ ′, σh′) = (gg′, τ ′, σh′),

if σh = σh′ and τ = τ ′. On the other hand, by (A.13) we have

F((g, σhτ, β(h,τ))·(g′, σh′τ ′, β(h′,τ ′))

)= F

(gg′, σh′τ ′, β(h′,τ ′)

)= (gg′, τ ′, σh′),

with σh = σh′ and τ = τ ′. Hence F preserves the composition law · .Concerning the composition law �, by (13) we have


)� F

(g′, σh′τ ′, β(h′,τ ′)

)= (g, τ, σh) � (g′, τ ′, σh′) = (gg′, τ ′, σhh′)

with the condition that σh′τ ′ = τ . On the other hand by (A.12) we have

F((g, τ, β(h,τ)) � (g′, τ ′, β(h′,τ ′))

)= F

(gg′, σhh′τ ′, β(hh′,τ ′)

)= (gg′, τ ′, σhh′),

with the condition σh′τ ′ = τ . This implies that F preserves the composition law �.Finally, concerning the composition law ×, by (13) we have

F(g, τ, β(h,τ)

)× F

(g′, τ ′, β(h′,τ ′)

)= (g, τ, σh)× (g′, τ ′, σh′) = (gg′, ττ ′, σhτσh′τ−1),

while by (A.11) we have

F((g, τ,β(h,τ))× (g′, τ ′, β(h′,τ ′))

)=

= F(gg′, σhτσh′τ ′, β(hτ(h′),ττ ′)

)= F

(gg′, σhτ(h′)ττ ′, β(hτ(h′),ττ ′)

)= (gg′, ττ ′, σhτ(h′)) = (gg′, ττ ′, σhτσh′τ−1).

Hence F preserves the composition law ×. Finally observe that

F(e, ι, β(e,ι)

)= (e, ι, ι),

F(e, τ, β(e,τ)

)= (e, τ, ι), τ ∈ Inn(G),

F(e, τ, β(h,τ)

)= (e, σhτ, σh), h ∈ G, τ ∈ Inn(G).

This means that F sends 0–, 1–, 2–arrows of I(2G) in 0–, 1–, 2–arrows of 3G respectively,completing the proof.


Note that the composition law · of 3G is Abelian. So the only inner ·–automorphism of3G is the identity automorphism. Hence the first two composition laws of the 4–categoryI(3G) coincide and are Abelian.

The form of a 3–cochain. We derive the formula (24) which defines a 3–cochain. Letx denote a 3–tuple (x1, x2, x3) of mappings, where xi : Σ1(K) → (3G)i, for i = 1, 2, 3.This amounts to saying that

x1(b) = (e, τb, ι), b ∈ Σ1(K),

x2(c) = (e, αc, γc), c ∈ Σ2(K),

x3(d) = (v(d), σd, ηd), d ∈ Σ3(K),

where τ : Σ1(K) → Inn(G), α, γ : Σ2(K) → Inn(G), σ, η : Σ3(K) → G, and v : Σ3(K) →Z(G). Such a map x is a 3–cochain whenever it satisfies equations (17) and (18). Byimposing that x satisfies these two equations we will arrive at formula (24). In the proofwe adopt the notation introduced in the proof of Lemma 3.3. Furthermore we will makeuse of the relations (1) and of the composition laws (13). Let us start by imposing thatx satisfies equation (17), that is

x2(c) � x1(∂1c) = x1(∂0c)× x1(∂2c) � x2(c),

for any 2–simplex c. The l.h.s. of this equation reads

x2(c) � x1(∂1c) = (e, α, γ) � (e, τ1, ι) = (e, τ1, γ)

if τ1 = α. Using this condition on the r.h.s. of equation (17) we have

x1(∂0c)× x1(∂2c) � x2(c) = (e, τ0, ι)× (e, τ2, ι) � (e, τ1, γ)= (e, τ0τ2, ι) � (e, τ1, γ)= (e, τ1, γ),

if γτ1 = τ0τ2, that is γ = τ0τ2τ−11 . Hence x satisfies the equation (17) whenever

x1(b) = (e, τb, ι), b ∈ Σ1(K),

x2(c) = (e, τ∂1c, τ∂0cτ∂2cτ−1∂1c), c ∈ Σ2(K)

In order to obtain the form of the component x3, we now impose that x satisfies theequation (18), that is

x3(d) · x1(∂01d)× x2(∂3d) � x2(∂1d) = x2(∂0d)× x1(∂23d) � x2(∂2d) · x3(d),


for any 3–simplex d. The l.h.s. of this equation reads

x3(d) · x1(∂01d)×x2(∂3d) � x2(∂1d) =

= (v, σ, η) · (e, τ01, ι)× (e, τ13, τ03τ23τ−113 ) � (e, τ11, τ01τ21τ

−111 )

= (v, σ, η) · (e, τ01τ13, τ01τ03τ23τ−113 τ−1

01 ) � (e, τ11, τ01τ21τ−111 )

= (v, σ, η) · (e, τ01τ13, τ01τ03τ23τ−113 τ−1

01 ) � (e, τ11, τ01τ13τ−111 )

= (v, σ, η) · (e, τ11, τ01τ03τ23τ−113 τ−1

01 τ01τ13τ−111 )

= (v, σ, η) · (e, τ12, τ01τ03τ23τ−111 )

= (v, σ, η) · (e, τ12, τ00τ20τ−110 τ02τ22τ

−112 )

= (v, σ, η) · (e, τ12, γ0γ2)

= (v, τ12, γ0γ2),

if σ = τ12 and η = γ0γ2 = τ00τ20τ−110 τ02τ22τ

−112 . Hence the l.h.s. of equation (18) determines

the form of the third component of x, that is

x3(d) = (v(d), τ∂12d, γ∂0dγ∂2d), d ∈ Σ3(K).

Hence x is equal to (24). What remains to be shown is that r.h.s. of (18) equals the l.h.s.,that is, it is equal to (v, τ12, γ0γ2). Since γ2τ12 = τ02τ22 = τ10τ23, we have

x2(∂0d)× x1(∂23d) � x2(∂2d) · x3(d) = (e, τ10, γ0)× (e, τ23, ι) � (e, τ12, γ2) · (v, τ12, γ0γ2)

= (e, τ10τ23, γ0) � (e, τ12, γ2) · (v, τ12, γ0γ2)

= (e, τ12, γ0γ2) · (v, τ12, γ0γ2)

= (v, τ11, γ0γ2),

completing the proof.

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Dipartimento di Matematica, Universita di Roma “Tor Vergata”Via della Ricerca Scientifica 1, 00133, Roma, ItalyEmail: [email protected]

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